We present an extended optical characterization of an adaptive microfluidic silicone-membrane lens at a wavelength of 633 nm, respectively 660 nm. Two different membrane variations; one with a homogeneous membrane thickness, and one with a shaped cross section, have been realized. This paper includes the theoretical predictions of the optical performance via FEM simulation and ray tracing, and a subsequent orientation dependent experimental analysis of the lens quality which is measured with an MTF setup and a Mach-Zehnder interferometer. The influence of the fabrication process on the optical performance is also characterized by the membrane deformation in the non-deflected state. The lens with the homogeneous membrane of 5 mm in diameter and an aperture of 2.5 mm indicates an almost orientation independent image quality of 117 linepairs/mm at a contrast of 50%. The shaped membrane lenses show a minimum wave front error of WFERMS=24 nm, and the lenses with a planar membrane of WFERMS=31 nm at an aperture of 2.125 mm.
© 2009 OSA
In the last view years the development of adaptive lenses increased drastically. In contrast to conventional systems with mechanical moving parts, adaptive lenses show no mechanical wear, however they are costly. In the meantime, three lenses based on three different working principles are commercially available. The lenses of Varioptics use electrowetting for tuning. These lenses, which have a clear aperture of up to 2.5 mm, have the potential to replace intricate conventional lens systems in consumer electronics (mobile phones, PDAs, laptops) . However, Optotune developed adaptive optical elements with a larger aperture of 6 mm for the application in digital cameras and in microscopes. The adaptive lens is based on electroactive polymers which deform an elastic lens, similar to that of the human eye . The commercial product of Holochip consists of a fluid-based singlet lens, where the interface is set by an elastic silicone membrane. The focal length of the lens with an aperture of 10 mm is tunable by the mechanical moving of the clamping ring . As an alternative to the three commercially available lenses we have recently presented an adaptive silicone membrane lens with an integrated piezo-actuator, which is suitable for use in optical systems with an aperture of 2.5 mm [4,5]. Thereby, we follow the membrane fluid lens concept and integrate a fast piezoelectric pump for actuation. Besides the papers on silicone lenses which have already been published [3,6,7] we present an extensive optical characterization of our adaptive silicone-membrane lens, which comprises the theoretical behavior and optical measurement results. We investigate especially the influence on gravity in horizontal and vertical lens orientation for lenses with homogeneous and inhomogeneous membrane thicknesses.
2. Design and fabrication
The system consists of two components: a lens and a pump actuator, which are connected via a glass substrate (see Fig. 1). The lens chamber is filled with water, and consists of a silicone membrane with a homogeneous or an inhomogeneous thickness, and of a supporting ring (see Fig. 2). An aluminum stabilisation ring is enclosed in the silicone ring to support the highly flexible structure and help to suppress distortions. The pump chamber, which is also filled with liquid, is made up of a piezo-bending actuator embedded in silicone. The exchange of liquid between the pump and lens chamber is enabled by orifices in the glass substrate. If a voltage is applied to the piezo-bending actuator, its interior bulges upwards so that the lens fluid is displaced from the pump chamber into the lens chamber. This causes a growing bulge of the lens membrane and thus a reduction in the focal length of the lens.
Both chambers (lens and actuator) are fabricated with a surface roughness of R a<40 nm by a polydimethylsiloxane (RTV 615 of GE Silicones) casting process in a hot embossing machine. During manufacture, a brass substrate produced by means of ultra-precise milling (UPM) serves as the negative form. In case of a homogenously thick membrane we control the membrane thickness t membr by spacers in the hot embossing machine. For the inhomogeneous membranes we place concave shaped lenses in the mould and control the residual thickness t resid, respectively, in a range of 20 to 1250 µm [4,5]. Generally, the volume shrinkage of the silicone is counteracted by the stamp force during the hardening process in the hot embossing machine. Since this force, however, is constant for all membrane thicknesses, the volume shrinkage for several membrane thicknesses increases with the cube with the membrane thickness.
After the fabrication of the subcomponents (glass substrate, lens and pump chambers) they are connected by means of oxygen plasma bonding . Afterwards, the lens is filled with water in a vacuum chamber.
3. Theoretical behavior
This chapter is devoted to the analysis of the theoretical opto-mechanical lens behaviour. In order to judge the lens quality, we examined the wave front error (WFERMS) of the different membrane shapes in a horizontal and a vertical lens orientation. First, a purely mechanical model of the lens chamber (membrane and supporting ring) is considered. In the simulation we approximate the aluminum stabilisation ring height over the whole lens chamber, due to the thin residual silicone layer thickness. The interconnection between the aluminum and the silicone surface is modeled by a fixed boundary condition at the outside of the lens chamber. The fixed boundary condition at the bottom side of the lens chamber models the connection to the glass substrate (see Fig. 3). For the silicone we set an elastic modulus of 1.54 MPa  and a Poisson ratio of 0.49.
The lens shape simulation in horizontal orientation is done by a rotationally symmetrical FE model with ANSYS Mechanical (see Fig. 3). The influence of the fluid behind the membrane is calculated with a hydrostatic pressure p hydr,h depending on the membrane amplitude and the membrane position. The simulation setup for the vertical lens orientation consists of a full 3D-model. In this case, the hydrostatic pressure p hydr,v points perpendicular to the membrane surface as a function of the membrane position. Second, after the mechanical simulation, the membrane shape (inner and outer side) is expressed in Zernike polynomials over 50% of the lens diameter , using software for mathematical analysis. For the focal length f lens and the wave front error in RMS (WFERMS) a water (refractive index: n liquid=1.33, n PDMS=1.4282, density: ρ liquid=1000 kg/m3) filled lens inclusive silicone-membrane is calculated by using software for ray tracing (ZEMAX). Figures 4 and 5 show the wave front error (WFERMS) as a function of the focal length flens for lenses with homogeneous membranes in horizontal and vertical orientation. Generally, in horizontal position the wave front error is decreasing for increasing focal length and decreasing membrane thickness. The influence of the membrane thickness is traced back to the reduction of the membrane stiffness which shifts the inflection point of the membrane in the direction of the membrane fixation, hence the spherical aberration of the lens decreases. In the horizontal orientation the hydrostatic pressure of the lens fluid increases the membranes radius of curvature and decreases the spherical aberration, which results in a rising lens performance.
Actually, the membrane of t membr=50 µm shows a minimum aberration of WFRRMS=0.02 λ at 72 mm focal length. This point indicates the change of sign in spherical aberration. In comparison to the horizontal orientation the simulation results of the lens in vertical show the opposite behaviour. Thereby, the wave front error is decreasing for increasing membrane thicknesses. Here, the reason can be traced back to the inhomogeneous hydrostatic pressure distribution which deforms the membrane asymmetrically. Thereby, the influence of the hydrostatic pressure is decreasing for increasing stiffness, respectively increasing thickness, of the membrane. The performance of the thickest membrane of 230 µm is almost independent on the lens orientation.
For a further visualisation of the effect Figs. 4 and 5 show two interferograms of the lens with a homogeneous membrane thickness of 150 µm at a focal length of 200 mm in horizontal and vertical orientation. On the one hand, the interferogram of the horizontal position illustrates only rotation symmetrical wave front errors dominated by the spherical aberration. On the other hand, in vertical position the coma aberration in z-direction is dominating due to the “belly” shape of the membrane.
Now, we will concentrate on the lenses with inhomogeneous membranes in horizontal orientation. Thereby, the influence of the residual membrane thickness t resid at 10 mm radius of curvature R membr is investigated while the hydrostatic pressure of the lens is neglected here (see Fig. 6). This is possible since the lens membrane is always thick and is not as drastically influenced by the gravity as was shown before.
As expected the curves show the smallest aberration in the non-deflected initial state at a focal length of 125 mm. For the deflected situations (increasing or decreasing focal length) the wave front error increases drastically with a decreasing residual membrane thickness t resid. Intuitively, for a smaller residual thickness we would expect a better hinge function, however, we observe a drastical increase of the wave front errors with deceasing thickness. In the deflected state, membranes with thicker residual membrane thicknesses are deforming more homogeneously in comparison to thinner ones, hence the wave front error is decreasing.
4 Experimental results
First, we qualify the fabrication process of the lens chambers. For this purpose we measured the deformation of the upper side of a homogeneous membrane directly after casting (without oxygen plasma bonding to the glass substrate) in a non-deflected state. We used a white light interferometer (Zygo new view 5000) in the stitching mode application. The lens chambers with homogenous membranes are scanned over the full membrane surface and the inhomogeneous one only with two perpendicular lines. Figure 7 shows the averaged measurement results of the homogeneous membranes with errors of two different lenses. All homogeneous membranes are sagging. The effect is increasing with increasing membrane thickness, due to the rising volume shrinkage.
Exemplarily, the radial symmetrical lens chamber is simulated with a volume shrinkage of 1.4% (see Fig. 8). As already shown in Fig. 3, the fixed boundary condition at the outside of the chamber indicates the interconnection to the aluminium stabilisation ring. However, the roll boundary condition at the bottom of the lens chamber models the contact surface during the measurement directly after the casting process. On the one hand, during the casting process the volume shrinkage of the silicone will stretch the membrane in radial direction due to the fixation at the aluminium stabilisation ring. On the other hand the shrinkage in z-direction deforms the supporting ring in the inner area, which, at the end, causes the sagging of the membrane.
For the inhomogeneous membranes, Fig. 9 shows the deformation of the upper membrane side in a similar way. Thereby, the residual membrane thickness t resid is varied while the radius of curvature of the casting mould of 9.3 mm is kept constant. In contrast to the homogeneous membranes, Fig. 9 shows a positive deflection of the upper membrane side. The amplitude increases for a decreasing residual membrane thickness. However, also in this case the deformation is caused by the shrinkage of the silicone during the hardening process. Due to the stretching in radial direction the inhomogeneous membrane bulges upwards. The reduction of the residual membrane layer reduces the membrane stiffness at the clamping, which increases the deformation of the membrane. In comparison to the theoretical behaviour without volume shrinkage, the measured focal length of the filled lenses in the initial state is thus reduced by 40%.
Second, we are focusing on the optical monochromatically characterisation of the lenses. The image quality, on the basis of the modulation transfer function (MTF), is analysed with a USAF 1951 test chart at a wave length around 660 nm. The variable parameters are the membrane thickness and the focal length. Thereby, the fluidic lenses are characterized in a parallel ray optical bench, which compensates for the negative focal length of plano-concave lenses. To simplify the lens handling of the plano-concave lenses a revolver stage is used, which contains eight lenses with focal length ranging from -1000 to -50 mm (see Fig. 10). The investigation of the lens orientation (horizontal and vertical) is realized by the rotation of the whole measurement setup .
Figures 10 and 11 show the resolution in linepairs/mm at 50% contrast as a function of the focal length in horizontal and vertical lens position. The lens diameter is 5 mm and the aperture 2.5 mm. The diagram of the lenses in horizontal lens position shows a maximum resolution of 117 lp/mm at a membrane thickness of 50 µm. It becomes clear that using thicker membranes the lens resolution will decrease. However in comparison to the theory, the image quality as a function of the membrane thickness shows no strict correlation which is related to the process tolerances. In vertical position the difference to the horizontal position is only minor. The resolution decreases slightly to 112 lp/mm due to the gravitational influences of the lens fluid. In comparison to the theoretical behavior the gravitational influence of the lens fluid is decreasing. The reason here is the process-related volume shrinkage, which increases the stiffness of the membrane.
For further characterization we use a Mach-Zehnder interferometrical setup to measure the wave front errors (WFRRMS) of the lenses at a wave length λ measure of 633 nm. The focal lengths of the adaptive lenses were adjusted by the exact distance variation between a laser objective and the adaptive lens. The detailed configuration of the interferometer is shown in . Figure 12 shows the measurement results of the lenses with homogenous membranes as a function of the focal length for diverse membrane thicknesses in horizontal lens orientation at an aperture of 2.125 mm.
The diagram shows a minimal wave front error WFERMS of 0.047 λ for a membrane thickness of 50 µm at a focal length of 800 mm. The measurement result of the thinnest membrane is in a good correlation with the image quality in Fig. 10. The results for the other membrane thicknesses differ due to the non-equal apertures used for the two measurement techniques. The solid line without measurement points in Fig. 12 shows the simulated wave front errors at a membrane thickness of 300 µm under consideration of the volume shrinkage which was discussed before. The comparison of the measured wave front error to the theoretical behavior inclusive volume shrinkage is in good correlation for short focal length (f lens <200 mm).
Figure 13 shows the wave front errors WFERMS as a function of the focal length for inhomogeneous membranes with diverse residual membrane thicknesses t resid at a mould radius of curvature of 9.3 mm. All curves show a minimal aberration at a focal length of 66 mm. The wave front error is increasing for decreasing membrane thickness and deflection of the membrane. In comparison to the homogeneous membrane the minimum wave front error WFERMS of the inhomogeneous membrane form could be reduced to 0.037 λ. The trend of the measurement results is in correlation to the simulated behavior which is plotted additionally for a residual membrane thickness of 250 µm inclusive process dependent volume shrinkage. The simulation and the measurement show the same curve shape, but with an offset in the focal length and the wave front error. The disagreement results mainly from fabrication tolerances of the lens chamber.
In this contribution we focused on the optical performance of an adaptive silicone lens system. The image quality was simulated based on a mechanical FE model with a subsequent membrane shape analysis which is expressed in Zernike polynomials. The process verification of the planar membranes showed a sagging, respectively a bulging in case of the shaped ones, due to the process related volume shrinkage.
Two measurement techniques (MTF-setup and Mach-Zehnder-Interferometer) have been used to verify the optical lens behavior. The resolution measurement showed an almost orientation independent optical imagine quality at a used lens diameter of 50%. The maximum resolution of 117 lp/mm was achieved for a homogenous membrane thickness of 50 µm at a focal length of 1000 mm.
We finally conclude that the question, whether to use an adaptive lens with a homogeneous membrane or with an inhomogeneous one, is application specific. On the one hand, lenses with planar membranes are reasonable for a constant optical lens quality (WFERMS=31…43 nm at d aperture=2.125 mm). On the other hand, the application of lenses with shaped membranes are reasonable for a higher optical lens quality (WFERMS=24…50 nm at d aperture=2.125 mm) at a smaller focal length range around a working point.
This project is supported by the German Research Foundation DFG in frame of the project EAGLE being part of the priority programme 1337 Active Microoptics.
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