Nonlinear large deformation of a transparent elastomer membrane under hydraulic pressure was analyzed to investigate its optical performance for a variable-focus liquid-filled membrane microlens. In most membrane microlenses, actuators control the hydraulic pressure of optical fluid so that the elastomer membrane together with the internal optical fluid changes its shape, which alters the light path of the microlens to adapt its optical power. A fluid-structure interaction simulation was performed to estimate the transient behavior of the microlens under the operation of electroactive polymer actuators, demonstrating that the viscosity of the optical fluid successfully stabilizes the fluctuations within a fairly short period of time during dynamic operations. Axisymmetric nonlinear plate theory was used to calculate the deformation profile of the membrane under hydrostatic pressure, with which optical characteristics of the membrane microlens were estimated. The effects of gravitation and viscoelastic behavior of the elastomer membrane on the optical performance of the membrane microlens were also evaluated with finite element analysis.
© 2014 Optical Society of America
Miniaturized adaptive optical systems have attracted a great deal of attention due to their potential applications in consumer electronics and biomedical diagnostic systems. Variable-focus liquid-filled membrane microlenses have been developed by several researchers as promising candidates for miniaturized adaptive optical systems [1–13]. The application of hydraulic pressure on optical fluid makes a transparent elastomer membrane deform, which tunes the focal length of a membrane microlens. Unlike conventional solid-state focus-tunable lens systems, variable-focus liquid-filled membrane microlenses cause several design issues including response time, precision control and spherical aberration. Furthermore, for hand-held electronics applications, variable-focus microlenses are expected to be small and to have low driving voltage, low power consumption, high thermal stability and low fabrication cost. Therefore, actuators to produce hydraulic pressure in liquid-filled microlenses are a crucial aspect of opto-mechanical performance. Several actuators such as electromagnetic actuators [2,10,11], piezo-bending actuators [3–6], relaxor ferroelectric polymer actuators [7,14], thermal actuators , and dielectric elastomer actuators  were developed and integrated into microfluidic membrane lenses.
In our previous work , we designed and fabricated a varifocal liquid-filled microlens operated by four poly(vinylidene fluoride-trifluoroethylene-chlorotrifluoroethylene) [P(VDF-TrFE-CTFE)] polymer actuators . The design and operation concept of the varifocal microlens developed by Choi et al.  is shown in Fig. 1. The main frame of the varifocal microlens is composed of three parts: a glass frame and two micromachined silicon frames. The lower silicon frame micromachined to have a lens hole and four actuator chambers was bonded to the glass frame. The top surface of the lower silicon frame was covered with transparent elastomer membrane so that the internal volume confined by the elastomer membrane, the lower silicon frame, and the glass frame is filled with optical fluid. Multilayered P(VDF-TrFE-CTFE) polymer actuators were fabricated on the upper silicon frame, and directly bonded onto the elastomer membrane on the lower silicon frame. Therefore, polymer actuators produce bending deformation under applied electric field, which pushes optical fluid in fluidic chambers into the lens part, which in turn changes the shape of the transparent elastomer membrane. Therefore, the shape change of the membrane tunes the focal length of the varifocal microlens.
For hand-held electronics applications, Choi et al.  and Pouydebasque et al.  designed and fabricated an array of varifocal liquid-filled microlenses on a wafer using microfabrication. This method is considered to be cost-effective for mass production and the fabricated varifocal microlenses are quite thin (less than 1 mm thick), which is desirable for mobile phone applications in particular. However, as the thickness of optical liquid confined in the microlens decreases, the response time of the liquid-filled microlens increases due to the viscosity of the optical fluid. Therefore, design of varifocal microlenses should be optimized to provide a fast response time with a small size.
Silicone elastomers such as polydimethylsiloxane (PDMS) are usually selected for optical membranes, since they have high optical transmittance and low Young’s modulus, and are capable of withstanding large deformation without permanent deformation. On the other hand, mechanical deformation of transparent optical membranes specifies the lens shape of liquid-filled microlenses, which results in several design issues with regard to optical performance such as effective aperture, dioptric power, optical transmittance, aberration, etc. First of all, the dynamic response of the membrane microlenses depends highly on the viscosity of the optical fluid and the viscoelastic properties of the optical membrane, in which the deformed shape of the membrane under hydraulic pressure is critical to the optical performance of the microlens.
In this paper, nonlinear large deformation of a transparent elastomer membrane under hydraulic pressure is analyzed and its effect on the optical performance of a variable-focus liquid-filled membrane microlens is investigated. The dynamic fluid-structure interaction (FSI) among the P(VDF-TrFE-CTFE) polymer actuators, optical fluid, and elastomer membrane in the liquid-filled varifocal microlens developed by Choi et al.  is analyzed in Sec. 2. The deformation of an elastomer membrane under hydrostatic pressure is analyzed by geometrically-nonlinear plate theory and its effects on the optical performance of the varifocal microlens are evaluated in Sec. 3. Using finite element analysis (FEA), Sec. 4 assesses the effect of gravity on spherical aberration of the varifocal microlens. Optical drift originating from the viscoelasticity of the elastomer membrane is analyzed with FEA in Sec. 5. Finally, Sec. 6 concludes this study.
2. Response time of microlenses: fluid-structure interaction analysis
The profile of a transparent elastomer membrane in liquid-filled membrane microlenses is controlled by hydraulic pressure generated by electroactive polymer actuators, and the dynamic deformation behavior of the membrane is crucial for opto-mechanical performance. For example, the shorter response time assures faster autofocus time of the microlenses. In this section, as a model problem, the dynamic fluid-structure interaction (FSI) among electroactive polymer actuators, optical fluid, and elastomer membrane in the liquid-filled varifocal microlens developed by Choi et al.  is analyzed with commercial software, COMSOL 3.3 . As illustrated in Fig. 1, four electroactive polymer actuators produce bending deformation to push optical fluid through the microfluidic channel into the center portion. The hydraulic pressure of optical fluid in turn causes the shape of the transparent elastomer membrane to change. The analysis model consists of a transparent elastomer membrane, optical fluid contained in a volume of 6 × 6 × 0.3 mm3, and four poly(vinylidene fluoride-trifluoroethylene-chlorotrifluoroethylene) [P(VDF-TrFE-CTFE)] polymer actuators as the electroactive polymer actuators as shown in Fig. 2(a). The height of the microfluidic channel, the diameter of the membrane, and the thickness of the membrane are thus considered to be 0.3 mm, 2.4 mm, and 50 μm, respectively. Young’s modulus, Poisson’s ratio, density, and thickness of the P(VDF-TrFE-CTFE) polymer actuators are set to be 400 MPa, 0.48, and 1.879 g/cc, and 50 μm, respectively. In FSI analysis, piezoelectric deformation of the P(VDF-TrFE-CTFE) polymer actuators was simulated by applying uniform pressure on the outer surface of the actuator, given as a function of time asFig. 2(b). The actuator pressure given in Eq. (1) was chosen to take into account the time delay of electromechanical response of the polymer actuator itself. A transparent elastomer membrane, polydimethylsiloxane (PDMS) is used, which has density, Young’s modulus, and Poisson’s ratio of 0.965 g/cc, 1.12 MPa, and 0.48, respectively . The density of the optical fluid was assumed to be 1.80 g/cc, while the viscosity of the optical fluid varied from 1 cP to 1000 cP. Therefore, the FSI simulation was performed for various values of the viscosity of optical fluid. The transient response of the deflection at the center of the membrane is plotted in Fig. 2(b), showing that the varifocal microlens is a dynamically over-damped system due to the viscosity of the optical fluid, and thus, no fluctuation was observed. The steady-state deflection of the membrane was determined to be 47.8 μm. The response time of the microlens was defined to be the elapsed time for the membrane deflection to reach 98% of its steady-state value, 47.8 μm. The response time was plotted as a function of the viscosity of optical fluid in Fig. 2(c), showing that the response time is approximately linear with the viscosity of optical fluid. Figure 2(c) also shows that when the viscosity of optical fluid is 100 cP, the response time is 17 ms. Choi et al.  used Fomblin PFPE M03 manufactured by Solvay S. A. as the optical fluid, which has viscosity and refractive index of 54 cP at 20°C and 1.29, respectively. Therefore, we expect that the response time of the varifocal microlens is less than 17 ms, which agrees well with the experimental results reported by Choi et al. . This level of response time is considered to be fast enough for autofocus function in mobile applications. It is also worth noting that if the height of the microfluidic channel increases, the response time decreases, which may be desirable for optical fluid with high viscosity.
3. Optical performance of nonlinear elastomer membrane
In the previous section, it was observed that the varifocal liquid-filled microlens is a dynamically over-damped system such that no fluctuation of the elastomer membrane takes place during dynamic operation. We therefore infer that static plate theory can be used to analyze the deformation profile of the elastomer membrane, which is essential for its optical design. In this section, the elastomer membrane under hydrostatic pressure is analyzed by geometrically-nonlinear plate theory, which provides excellent estimation of its deformation profile unless any transient response within 20 ms is under consideration.
3.1 Deformation profile of an optical membrane under hydrostatic pressure
Let us assume that a uniformly loaded circular membrane with a clamped edge undergoes axisymmetric large deformation as depicted in Fig. 3(a). No surface tension, residual stress, or dynamic effect is considered here. Then, the governing equation becomes a system of nonlinear differential equations as follows :
Details of the solution technique of governing Eqs. (2) and (3) with the boundary conditions given in Eqs. (4) and (5) are given by Timoshenko and Woinowsky-Krieger , and thus we only bring their results here for compactness. Equations (2) and (3) can be solved by assumingEqs. (2) and (3) together with boundary conditions given in Eqs. (4) and (5), are automatically satisfied if the unknown coefficients and satisfy the following recursive relations:
Numerical results for the deformation profile of an elastomer membrane are shown in Fig. 3(b), for which a Young’s modulus, Poisson’s ratio, thickness, and radius of PDMS membrane of 1.12 MPa, 0.48, 50 μm, and 1,200 μm, respectively, were used. Near the clamped edge, the deformation profile largely deviates from a spherical shape due to the boundary constriction, which restricts a change of curvature in the membrane, and thus, forces the edge region to be discarded as an optical lens. Figure 3(c) displays the maximum deflection at the center of the optical membrane as a function of normalized pressure for several thicknesses of the optical membrane. As the thickness of the optical membrane increases, the maximum deflection at the center of the optical membrane becomes linearly proportional to the normalized pressure.
3.2 Optical performance of aspherically-deformed elastomer membrane
For optical analysis, the deformation profile of an elastomer membrane given in Eq. (6) can be converted into the following function:Eq. (11), is expanded in a power series, the convergence of the power series is not guaranteed . The deformation profile given in Eq. (11) with vertex curvature , conic constant , and higher aspheric coefficients determined in terms of turns out to be valid only near the center, and not near the edge of the membrane. Therefore, without loss of generality, the conic constant is assumed to be (paraboloid), and thus, the vertex curvature and the higher aspheric coefficients are determined in terms of asEq. (12) is plotted in terms of normalized pressure for various membrane thicknesses in Fig. 4. When , the vertex curvature dramatically changes at the low pressure level (< 1). As the thickness of the optical membrane increases, the vertex curvature of the optical membrane becomes linearly proportional to the normalized pressure. This tendency may suggest that the focal length of the varifocal microlens can be easily controlled for a thicker membrane.
The ray tracing simulation of the variable-focus liquid-filled microlens as shown in Fig. 1 was performed by using commercial software, Code V, in order to calculate the focal length and wave front error (WFE). The profile given in Eq. (11) with and given in Eq. (12) and with serves as the axisymmetric profile of the optical membrane made of PDMS ( = 1.38) in the optical analysis. Fomblin PFPE M03 manufactured by Solvay S. A. was used as the optical fluid, whose initial thickness and refractive index are = 300 μm and = 1.29, respectively . The thickness and refractive index of the glass frame were assumed to be = 300 μm and = 1.52, respectively. The aperture stop of the microlens was assumed to be 2.0 or 1.6 mm in the optical simulation. The ray tracing simulation showed that the membrane deflection, 47.8 μm, obtained from the FSI analysis in Sec. 2 corresponds to the change of focal length from infinity to 38.3 mm (aperture stop = 2.0 mm) and 34.5 mm (aperture stop = 1.6 mm), representing the change of optical power by more than 25 diopters. The root-mean-square (RMS) WFE (WFERMS) was plotted as a function of focal length for various values of the membrane thickness in Fig. 5. As the focal length increases, the WFERMS gradually decreases. For a thicker optical membrane, the WFERMS is slightly greater. It is worth noting that the WFERMS for the 1.6 mm aperture stop is much smaller than that for the 2.0 mm aperture stop. This difference implies that the edge effect due to the bending stiffness tends to increase spherical aberration.
4. Spherical aberration due to gravity
The membrane shape of variable-focus liquid-filled microlenses is influenced by gravitational force, since the membrane surface may be positioned parallel to the gravitational force and the hydrostatic pressure varies along the direction of the gravitation force. The gravitational effect on the membrane deformation was simulated with a commercial finite element analysis (FEA) program, ABAQUS Version 6.12 , in which the optical membrane is located in the xy plane, and the gravitational force is assumed to act along the y direction. Therefore, the hydrostatic pressure is given asFigure 6(a) shows a contour plot of the deformed shape of the elastomer membrane with a thickness of 50 μm under a hydraulic pressure = 2 Pa. It can be inferred from Fig. 6(a) that the linear distribution of the hydrostatic pressure caused undulation of the elastomer membrane in the y direction. Figure 6(b) represents the deformation profile along the vertical center line (y axis) of the elastomer membrane for thicknesses of 50 and 70 μm for various values of applied pressure .
The deformed surface profiles of the elastomer membrane obtained above by FEA were used in the optical simulation with commercial software, Code V, with which the effect of gravity on the optical performance (focal length and WFE) of the variable-focus liquid-filled microlens as shown in Fig. 1 was investigated. The root-mean-square (RMS) WFE (WFERMS) was plotted as a function of focal length for various values of the membrane thickness in Fig. 7. It is worth noting that since a thinner optical membrane produces larger deflection under a given hydrostatic pressure as shown in Fig. 6(b), the WFERMS due to gravity becomes greater for a thinner optical membrane as can be observed in Fig. 7 for large focal length (greater than 50 mm). On the other hand, as the focal length decreases (in other words, as the deformation of the optical membrane increases), the WFERMS due to gravity decreases due to the geometrical nonlinearity of the optical membrane, while the WFERMS due to the aspherically-deformed membrane increases as shown in Figs. 5 and 7. Therefore, compared to the WFERMS due to gravity, the WFERMS due to the aspherically-deformed membrane becomes dominant for small focal length (smaller than 50 mm). When the WFERMS in Fig. 7 is compared with that in Fig. 5, the WFERMS due to gravity seems to be negligible for the optical membrane of thickness 90, 110, and 130 μm. However, the optical membrane of thickness 50 μm shows dramatic increase of the WFERMS in the range of 50-100 mm focal length, meaning that the optical membrane of thickness 50 μm should not be used without applied pressure to prevent excessive WFE due to gravity. Therefore, it can be inferred that a thicker elastomer membrane reduces WFE due to the gravity effect. It is worth recalling that in order to reduce the WFE due to an aspherically-deformed membrane, the thickness of the elastomer membrane needs to be decreased. Therefore, the thickness of the elastomer membrane must be optimized to minimize the total WFE due to both the aspherically-deformed membrane and gravity.
In order to reduce spherical aberrations or WFE of a liquid-filled microlens, it is important to find the ranges of design parameters for which deformation of an elastomer membrane under hydrostatic pressure does not exceed a given limit. The hydrostatic pressure by the gravitational force can be given as and the coordinate has the order of the membrane diameter, . Therefore, the hydrostatic pressure due to gravitational force can be assumed to be for a parametric study. Let us consider the deformation of a linear elastic circular membrane under a uniform pressure , for which the maximum deflection at the center of the membrane can be given as Equation (15) provides us a design guideline for liquid-filled microlenses: All the parameters, in Eq. (15) must be selected to make a shape deviation less than a predetermined limit. It is worth noting that the diameter and thickness of the membrane are key parameters controlling the effects of gravity, since the shape deviation is proportional to the fifth power of the membrane diameter and inversely proportional to the third power of the membrane thickness.
5. Optical drift due to viscoelasticity of the elastomer membrane
Silicone elastomers like PDMS, typical optical membrane materials for liquid-filled microlenses, exhibit viscoelastic behavior. The viscoelasticity of silicone elastomers makes the opto-mechanical behavior of the membrane microlenses time-dependent. In particular, creep behavior (gradual change of deformation over a relatively long period of time under applied stresses) of the elastomer membrane may cause optical drift of liquid-filled membrane microlenses, for example, gradual change of focal length as a function of time. The viscoelastic behavior of an elastomer membrane was analyzed by a finite element method with a commercial program, ABAQUS Version 6.12 . Let us assume that the uniformly loaded circular elastomer membrane with a clamped edge undergoes axisymmetric deformation as depicted in Fig. 3(a). However, the elastomer membrane is considered to be a viscoelastic PDMS, for which the relaxation modulus is given as 16], which is a typical assumption in viscoelasticity . The uniform pressure was applied at = 0 s, that is, , where is a Heaviside step function. Figure 8(a) represents the deformation profile of the PDMS membrane under the uniform pressure obtained by FEA. As time increases, the deformation of the PDMS membrane gradually increases. The maximum deflection at the center of the viscoelastic PDMS membrane is denoted by , and the relative deflection at the center of the membrane is defined to be . If the normalized relative deflection, , at the center of the membrane is plotted as a function of time, the creep behavior of the membrane can be unified into a single curve as shown in Fig. 8(b).
The creep behavior illustrated in Figs. 8(a) and 8(b) may cause optical drift of liquid-filled membrane microlenses. For example, the focal length of a microlens may gradually change due to the viscoelastic behavior of the viscoelastic optical membrane as a function of time, which must be compensated to provide continuous quality of imaging. Figure 9 shows the percent change of the focal length of the liquid-filled membrane microlens for 10 seconds due to the viscoelastic behavior of the PDMS membrane. For focal lengths greater than 150 mm, the percent change in focal length over 10 seconds is fairly uniform, while for focal length less than 150 mm, the percent change in focal length over 10 seconds dramatically decreases as the hydrostatic pressure increases and thus the focal length decreases. This observation implies that when relatively high pressure is applied, creep behavior of the optical membrane and thus the optical drift of the liquid-filled membrane microlens diminish due to the geometrical nonlinearity of the optical membrane.
Nonlinear large deformation of a transparent elastomer membrane in a variable-focus liquid-filled membrane microlens was analyzed to study its optical performance. In most membrane microlenses, actuators control hydraulic pressure of the optical fluid so that the elastomer membrane together with the internal optical fluid changes their shape, which alters the light path of the microlens to adapt its optical power. Fluid-structure interaction simulation demonstrated that the viscosity of the fluid successfully stabilizes the fluctuations during dynamic operations, and the lens changes the optical power by more than 25 diopters within 20 ms. Geometrically-nonlinear plate theory was used to calculate the deformation profile of the elastomer membrane under hydrostatic pressure, with which optical characteristics of the membrane microlens were estimated. In order to estimate the effect of gravity on the optical performance (focal length and WFE) of the variable-focus liquid-filled microlens, FE analysis and optical simulation were performed. It is worth noting that the diameter and thickness of the membrane are the key parameters to control the effect of gravity, since the shape deviation is proportional to the fifth power of the membrane diameter and inversely proportional to the third power of the membrane thickness. The effect of the viscoelastic behavior of the elastomer membrane on the optical performance of the membrane microlens was also evaluated with FEA. The percent change in focal length over 10 seconds was approximately 7%.
This work was supported by the 2012 Research Fund of the University of Ulsan.
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