Peter Liebetraut,* Sebastian Petsch, Wolfgang Mönch, and Hans ZappeLaboratory for Micro-optics, Department of Microsystems Engineering—IMTEK, University of Freiburg,
Whereas macroscopic lenses and lens systems typi-cally use mechanical movement to tune focal lengthor magnification, microlenses may use a variety ofnovel mechanisms to achieve tunability. Electrowet-ting has been extensively developed for actuation oflenses based on liquid droplets [1], as has the use ofliquid crystals, both as an optical element itself [2,3]and for actuation of liquid lenses [4]. The use ofdistensible membranes, usually made from polydi-methylsiloxane (PDMS) and suspended over liquid-filled microcavities to act as tunable microlenses,is likewise well-developed [5,6] and has been appliedto lenses with diameters up to 20mm [7]. Most mem-brane lenses are actuated by pressure, generated, forexample, by on-chip piezoelectric actuators [8] or byinduced polymer swelling [9].
These concepts have some attendant disadvan-tages, including susceptibility to gravitational forcesand hence orientational sensitivity for liquid andmembrane lenses, or polarization-dependence andundesired birefringence in liquid crystal structures.Those drawbacks may be circumvented by using
flexible, solid-body elastomeric lens structures. Suchsolid-body lenses can be tuned in focal length by acontrolled deformation of their shape; this effecthas been demonstrated using thermal actuation[10] which is, however, slow. In this paper, we showthat solid-body elastomer lenses can be tuned withfast response by using magnetic and mechanicalactuation.
There are several benefits of bulk elastomerlenses. The material itself provides some advantagesover liquid lenses, since the reliable sealing of the op-tical liquids and electric insulation are of no concern.These lenses are also not prone to gravitational sag-ging and less susceptible to vibrational disturbances.Furthermore they are, in contrast to glass lenses,shock-resistant.
In contrast to membrane lenses and similarly toglass lenses, these bulk elastomer lenses come witha predefined surface in the nonactuated state, contri-buting a definite optical power to the total opticalpower of a multicomponent lens system. The focallength tuning range, about 10%, is sufficient to refo-cus at different object distances, while the main re-fractive power originates from additional lenses withfixed focal lengths. The concept is similar to themammalian eye, where the predominant refractive
power is provided by the cornea and the actual focus-ing is done by an elastomeric polypeptide lens.
2. Concept and Design
The design of the biconvex bulk elastomer lens isshown in Fig. 1. The lens consists of an elastomericmaterial (PDMS) and has a diameter d0 of 20mm.Embedded in the PDMS lens are four metal anchors;both curved and straight anchors were investigated,as sketched in Fig. 1. Since all mirror planes andsymmetry axes are preserved, even under strain,the lens always shows D4h symmetry (point groupin Schönflies notation [11]). The anchors also definethe clear aperture, a0 ¼ 16mm, which is the distancebetween the two vertices of the opaque anchors atzero strain.
The focal length of this elastomeric lensmay be var-ied by application of strain. When a force, Fmag, is ap-plied radially to the anchors, these move outwardwhile the counterforce (Felast) is provided by theYoung’s modulus, E, of the polymer (E ¼ 0:08MPa).While straining the lens in the equatorial plane,the total volume of the lens remains constant (Pois-son’s ratio μ ¼ 0:5 is generally assumed for PDMS),hence the radius of curvature increases and so doesthe focal length of the lens. The strain, ϵ, which usesthe initial clear aperture a0 of the lens as a reference,is expressed as
ϵ ¼ Δaa0
¼ Δa16mm
; ð1Þ
where Δa is the diameter increment due to themovement of the anchors.
3. Fabrication
The biconvex solid-body elastomer lenses are fabri-cated using two commercial concave glass lenses asmaster structures in a reaction injectionmolding pro-cess. Two types of lenses are used, which are 20mm indiameter and have focal lengths of −30mm(R0 ¼ 23:54mm) and −50mm (R0 ¼ 25:84mm), re-spectively. The glass lenses are coated with a thinfluoropolymer layer and inserted into a tubular mold.The fluoropolymer coating facilitates demolding ofthe elastomer replica and is applied by spin-coatinga solution of 1:5wt:% AF-1600 (DuPont, USA) inFC-40 (3M, USA). Four metallic laser-patternedanchors (thickness, 0:5mm; height, 2mm; width,
8mm) are aligned symmetrically in two diametricalpairs, 2mm from the edge of the lens.
The tubular mold has four small openings in theequatorial plane of the lens to be fabricated, whichare used to align the anchors properly. These open-ings are then sealed with plugs made of RTV 23(Altropol, Germany) to prevent the silicone precursorfrom leaking. The master lenses fit the mold tightly,hence no additional sealing is necessary.
The cavity is then filled with the liquid precursor ofa silicone elastomer (SE 17040, Dow Corning, USA),and cured for 120 min at 90 °C in an oven. Figure 2illustrates the process.
After the initial curing process, the silicone lenseswith incorporated anchors are dipped into a Sylgard184 (Dow Corning, USA)-n-Heptane solution(10∶1wt:=wt:) and are subjected to a final curingstep, resulting in a thin film that improves the hand-ling properties of the lens (particularly reducing thesusceptibility to dust particles), with negligible im-pact on mechanical and optical properties. Table 1shows an overview of the initial properties of the fab-ricated lenses.
4. Modeling and Simulation
We use Comsol Multiphysics to obtain the lensprofile by finite element (FEM) simulation. Sincethe four discrete anchors are represented by D4hsymmetry, only one-eighth of the lens needs to be si-mulated, which greatly reduces demands on comput-ing. The profile of the entire lens can be generated bypostprocessing.
Table 1. Initial Properties of Fabricated Bulk Elastomeric Lenses
Fig. 1. (Color online) Schematic sketch of the lens geometryviewed along the optical axis. Strain is applied via four anchorsthat are (a) straight (T-shaped) or (b) curved (U-shaped).
Fig. 2. (Color online) Illustration of the lens fabrication process.(a) Positioning the first planoconcave master lens in a tubularmold and inserting the metallic anchors, (b) sealing the mold withsecond glassmaster lens and alignment of the anchors, (c) injectingPDMS and curing, (d) demolding.
The slope of the focal length versus strain curve,which is derived from the simulation data, varies tosome degree with the boundary conditions of the si-mulation, and this variation is of the same order ofmagnitude as the error due to the uncertainty of therefractive index, as shown in Fig. 3. This justifies lim-iting the curve fitting to a two-dimensional cross sec-tion of the lens, which comes with a significantreduction in computation time.
From the raw profile data, the focal length as afunction of strain may be easily calculated. After fit-ting a circle to the cross section of the simulated lensprofile to obtain the radius of curvature, we use theparaxial approximation to predict the focal length ofthe biconvex elastomeric lens
ΦðϵÞ ¼ 1f ðϵÞ ¼
�nL − n0
n0
�·�
2RðϵÞ þ
ðnL − n0Þ · tðϵÞnLRðϵÞ2
�;
ð2Þ
with ΦðϵÞ the refractive power, f ðϵÞ the focal length,RðϵÞ the radius of curvature, nL the refractive indexof the elastic lens (nL ¼ 1:408 ¼ nPDMS for λ ¼589nm) and n0 the refractive index of air. The lensthickness at the optical axis, tðϵÞ, is also variableand reduces under strain, since the total volume ofthe lens is preserved.
Figure 4 shows the surface of a lens with an initialradius of curvature of R0 ¼ 25:84mm at 10% strainas obtained by FEM simulation. At 10% strain, thecontour line at 1:4mm absolute height shows a non-uniform deformation at the edge of the lens due tothe embedded anchors. In the immediate surround-ings of the anchors, the lens suffers from aberrationsdue to the anisotropic stress distribution, whichcreates warps in these areas. When approachingthe center of the lens, the stress distribution becomes
more uniform and these deformations are increas-ingly leveled out, due to the high elasticity of the lensbody.
While the clear aperture is limited by the nontran-sparent anchors, we also define an effective aperturewith aeff ¼ 14mm, as indicated by the yellow circle inFig. 4, inside which the rotationally symmetric sphe-rical shape of the lens is satisfactorily preserved andaberrations are expected to be low. Thus, only theprofile data inside this effective aperture was usedto calculate the focal length from simulation dataand, in the experimental verification, the effectiveaperture was implemented by placing an aperturestop close to the lens body.
These simulations showed that the focal length de-pends primarily on the total applied strain and toonly a lesser extent on the direction and numberof points in which strain is applied. This effect is seenin Fig. 5(a) for two different lenses (R0 ¼ 23:54mmand R0 ¼ 25:84mm), where strain was applied tetra-gonally (all four anchors pulled uniformly) and bidir-ectionally (only two adjacent anchors pulled, e. g., at0° and 90°). Figure 5(a) shows the simulation resultsof focal length versus strain, indicating that the twodeformation modes, tetragonal and bidirectional,result in differing focal length only for strain valuesin excess of about 8%.
An additional numerical simulation of a lens witheight anchors (eightfold symmetry, D8h) providedfurther support for this thesis, verifying that thetuned focal range at small strain values depends onlyon the total strain, and not on its direction or numberof application points. This somewhat unexpected buthighly promising result implies that the number ofanchors can be kept to a minimum without loss offocal length tuning performance.
Fig. 3. (Color online) Comparison of the measured focal lengthtuning (triangles) with FEM simulation (bullets). The purpleshaded areas represent the level of uncertainty in the simulationprediction which would result, for example, from an uncertainty inthe refractive index of 5‰. The orange and red shaded areasrepresent the calculated experimental error.
Fig. 4. (Color online) Surface profile for a lens with R0 ¼25:84mm at 10% tetragonal strain, as obtained by FEM simula-tion. The contour lines are evenly spaced at 0:4mm pitch. Allscales are in millimeters and refer to the upper half of the lens,i. e., the total biconvex lens thickness at each point is twice thedisplayed value. The yellow circle indicates the effective aperture(aeff ¼ 14mm) of the lens, inside which the aberrations areexpected to be low.
A possible explanation for this effect is that the de-formation properties of the polymer are such thatmost of the induced strain remains at the peripheryof the lens. As a result, we expect that small asym-metries in actuation force on different anchors willnot noticeably affect the tuning behavior of the lens.We have, however, only considered focal length as avariable parameter; we will discuss the effect of dis-crete and asymmetric actuation on higher-order lensaberrations below.
5. Measurements
To verify the simulation results, the focal length ofthe lens was determined by measuring the distancebetween an object, its image, and the center of thelens for various levels of applied strain. Leavingthe image plane fixed, the focal length was deter-mined by shifting the object, a transparent black-and-white test slide, along the optical axis using amicrometer. The test slide was back-illuminated withan LED white light source, equipped with a colorfilter (λmax ¼ 540nm) to suppress chromatic aberra-tions and to facilitate the evaluation of the imagesharpness.
An aperture stop was used to define the effectiveaperture, as discussed above, and thus to suppressstray light and reduce distortion resulting fromthe presence of the embedded anchors at the periph-ery of the lens. The aperture stop also ensured thatthe pupil diameter remained constant as the anchorsmoved outward. The stop had a diameter of 8mm,thus resulting in initial f -numbers of f =# ¼ 4:1
and f =# ¼ 3:8 for elastomeric lenses with initial focallengths of 32:9mm and 30:2mm, respectively.
Several alternative actuation mechanisms wereused to mechanically deform the lens in the samemanner as in the simulation (tetragonal strain, ac-tuating all four anchors simultaneously, and bidirec-tional strain, using only two adjacent anchors). In afirst experiment, micrometers were used to apply thestrain. The configuration employing bidirectionalstrain was carried out with force sensors fixed totwo adjacent anchors (e. g., 180° and 270° positions)and the force–strain characteristic obtained by thisexperiment was later used to control the drive cur-rent of the magnetic actuators.
The measurements are shown in Fig. 5(b), andthey confirm the simulation results. The slope ofthe curves for tetragonal and bidirectional pullingare almost identical, yet the simulation predicts aslightly steeper slope than subsequently experimen-tally observed (see Figs. 3 and 6). Because of highstress at the periphery of the lens while actuatingasymmetrically, the strain was limited to approxi-mately 6% to prevent the lens from damage. Overallthe results, obtained in the control experiment withmechanical actuated lenses, confirm nicely the linearcorrelation between focal length tuning range andstrain.
The lens was subsequently deformed by applyingbiaxial, bidirectional strain, using electromagnetsto pull on the embedded anchors. Two adjacent an-chors (at 0° and 90°) were attached to the actuators,and the opposing anchors (180° and 270°, respec-tively) were fixed to force sensors. A maximum forceof about 600mN was applied, and the lens shapevaried accordingly. The lens responded withoutnoticeable delay to the applied magnetic force, andthe strain was adjusted using the previously ob-tained force–strain characteristic. Mechanical stopswere used to prevent excessive and irreversible de-formation of the lens. As predicted by FEM simula-tion, the optical axis of the lens shifted slightly in thedirection of actuation, such that a precision mechan-ical movement was used to realign the optical systemas the focal length was changed.
The experiment with electromagnetic actuation in-cluded measurements for four different lenses withtwo different initial focal lengths, equipped witheither straight (T-shaped) or curved (U-shaped) an-chors, to investigate the impact of the anchor shapeon focal length tuning; the typical characteristics,plotted as focal length as a function of electromagnetcurrent, can be seen in Fig. 6. The deviation ofmeasurement #41 can be attributed to a misalignedanchor, and the shape of the focal length-strain-characteristic could be reproduced over severalmeasurements.
Based on the force–strain characteristic obtainedin previous experiments, the focal range tuning asa function of strain rather than drive current couldalso be evaluated. A comparison of measured and si-mulated variation of focal length versus strain for
Fig. 5. (Color online) Change in focal length for tetragonal(symmetric), squares; and bidirectional (asymmetric), diamonds,strain. (a) Values derived from FEM simulation, (b) experimentalresults. Both graphs depict data for two lenses with initial focallengths of 30mm and 33mm. The dashed lines in (b) representa linear fit to the data. For the measurements, the error barsare derived from propagation of uncertainty.
lenses with R0 ¼23:54mm is shown in Fig. 3, verify-ing the linear correlation between strain and focallength as predicted by FEM simulation. Table 2 sum-marizes the maximum focal length tuning rangeobtained using magnetic actuation.
No significant difference in tuning range gener-ated by the two anchor types was observed, althoughthe anchor shape has an impact on the strain distri-bution inside the lens, as shown schematically inFig. 7. The curved anchors, though concentrically fol-lowing the outer shape of the lens, induce strain andstress peaks at their tips. Surprisingly, the experi-mental results show that the impact of these canbe neglected, as we saw in the comparison of Fig. 3.It is important to remember that both anchor typesare represented by the same point group, and hencethere is no change in the symmetry of actuation. Anydifference, if detectable, in focal range tuning be-tween the two anchor types would therefore havebeen a result of stress distribution due to the anchorshapes (and not the symmetry).
In addition to focal length, the impact of strain onimage formation is also of interest. Figure 8 showsthe image quality of the lens under different levels oftotal strain. Figure 8(a) depicts the original imagedobject, and in Figs. 8(b)–8(d), the images of the objectas formed by the lens at 0%, 5%, and 10% strain, re-spectively, are seen. Each image represents an image
of 896 × 896 pixels, and the edge length of each objectsquare is 1mm.
The object distance so and image distance si werechosen such that the lateral magnification is approxi-mately 2× (so ¼ 49mm, si ¼ 100mm) for the initialrelaxed state of the lens. The lateral magnificationincreases from 2.0 at 0% strain to 2.9 at 10% strain.
Because of dust on the lens surface and the inher-ent scattering of polymers with low glass transitiontemperatures, the image through the lens is subjectto reduced contrast, even in the relaxed state.
Since the lens thickness also affects the overallscattering of the lens body, the image contrast mightbe increased by reducing the lens thickness in futureexperiments.
Although the strain was applied asymmetrically,one can see in Figs. 8(c) and 8(d) that the image
Fig. 6. (Color online) Measured focal length as a function of elec-tromagnet actuation current for bulk elastomeric lenses with mag-netic bidirectional actuation. Results for both straight and curvedanchors are shown and show the typical hyperbolic characteristicof magnetic pull actuation.
Table 2. Focal Length Tuning Ranges Obtained by Experiment forApprox. 10% Strain for Lenses with Straight (T-shaped) and Curved
(U-shaped) Anchors
LensStrain(%)
Force(mN)
Change in FocalLength (%) (mm)
R25 T 9.7 408.3 10:5� 1:5 3.5R25 U 10.2 484.5 9:2� 1:5 3.0R23 T 9.0 392.6 9:5� 1:7 2.9R23 U 10.5 363.8 10:2� 1:7 3.1
Fig. 7. (Color online) Schematic representation of the strain dis-tribution inside the lens body for straight and curved anchors foruniaxially applied strain. Since the configuration is symmetricabout the vertical axis, only one-half of the lens is sketched, withσ representing the symmetry plane. In case of the straight anchors,the strain distribution is uniform, while in case of the curvedanchors, it peaks at the anchor edges.
Fig. 8. Images through the lens taken by a CCD camera (Stingray146B, Allied Vision Technologies GmbH, Germany) using incoher-ent Köhler illumination with a broadband white light source.(a) Object, (b) image at 0% strain, (c) image at 5% strain, (d) imageat 10% strain. The image plane and the field stop are slightly outof plane, resulting in a blurred image of the field stop [visible in(a) and (b)]. All subfigures have the same scale. The lateral mag-nification is approximately 2× with object distance, so ¼ 49mm,and image distance si ¼ 100mm for the lens in the relaxed statewith f 0 ¼ 32:8mm.
distortion remains relatively low, yet a small reduc-tion in image contrast can be observed at higherstrain, due to increased aberrations.
6. Discussion
Figure 3 compares the focal length tuning of the so-lid-body lenses, as derived from FEM simulation andmeasurement, in more detail. The results for lenseswith an initial radius of curvature of R0 ¼ 23:54mmare shown, and the experimental results are cor-rected for the characteristic of the magnetic actuator.The shaded areas indicate the ranges of uncertainty,which, in the case of the simulation data, is due to anassumed uncertainty of 5‰ of the refractive indexof PDMS.
The data depicted in Fig. 3 reveal an approxi-mately linear relationship between the focal lengthf and the applied strain ϵ, with a slope close to unity,if plotted in relative units (i. e., 10% strain results ina change in focal length of approximately 10%). TheFEM model shows a slightly larger change in focallength, around 12%, for this strain value.
One should be aware that the exact slope predictedby the simulation depends on a variety of simulationboundary conditions, such as meshing parameters,the precise value of the Poisson’s ratio for PDMS(taken to be μ ¼ 0:5) and also on parameters intro-duced in the postprocessing of the simulation data,e. g., refractive index and curve fitting parameters.We found that the effects of the simulation and post-processing parameters on the focal length tunabilityall vary in the same range as the error introduced bythe uncertainty of the refractive index (which isshown in Fig. 3).
FEM simulation is unlikely to produce reliablequantitative predictions concerning aberrations,yet the FEM model still provides valuable insightinto the behavior of the lens under strain.
It appears that most of the stress is distributed be-tween the anchors in the periphery of the lens, result-ing in the azimuthally homogeneous deformationbehavior that is observed within the effective aper-ture of the lens. With respect to a minimization oflens aberrations, employing strain with perfect ra-dial symmetry (D∞h) would be preferable; however,its realization remains technically challenging.
The FEM simulation also allows us to make rea-sonable predictions for the optical properties of thelens, as confirmed by experiment: the tuning effi-ciency (Δf for a given strain) is not as dependenton the symmetry as one might expect. As long as dis-crete anchors are used, the total amount of employedstrain governs the focal range tuning rather thansymmetry (with the exception of ideal D∞h symme-try). Using a larger number of discrete anchors wouldincrease the tuning efficiency at a given displace-ment of each anchor, or, in turn, would allow usinga smaller absolute displacement of the individualanchors to generate a predetermined strain. Higherdiscrete symmetry, therefore, also favors the reduc-tion of aberrations since it allows more uniform focal
length tuning at lower absolute displacements andhence causes less lateral contraction between the an-chors. This approach may be used to improve the per-formance of these bulk elastomeric lenses, albeit theactuation will then become more complex.
As an example of a more complex design, Fig. 9shows an image of an advanced and miniaturized ac-tuator with eightfold symmetry (D8h). The actuatorfits in standard Linos optical rail mounts and strainsan elastomeric lens symmetrically along all four axessimultaneously. The buried anchors can be adjustedindividually to prevent prestrain.
7. Conclusion
The deformation and optical behavior of magneti-cally actuated solid-body elastomer lenses has beenstudied by simulation and experiment. The variationof focal length with strain applied by embedded an-chors was seen to be linear over a range of 10%, andthe effect of asymmetric actuation on the focal rangetunability was seen to be minimal. In addition, theanchor shape has negligible effect on lens deforma-tion at the optical axis. These results provide usefulguidelines for the design of solid-body lenses andtheir actuation, as well as first indications of theoptical performance of these devices.
This work was funded by the German ScienceFoundation (DFG) within the framework of PriorityProgram 1337, “Aktive Mikrooptik”. We thank Y. Sunfor providing the model depicted in Fig. 9.
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