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Liquid lens offers a simple solution to achieve tunable optical powers. This approach, however, suffers from deteriorated resolution at high diopters. In this study, a plano-convex liquid lens with aspherical cross-section is developed. Such configuration allows for the lens profiles at high diopters to be close to spherical shapes by alleviating the edge-clamping effects. Resolution tests of a 6mm lens with optimized asphericity exhibit improved resolutions in both center and peripheral regions at 40 and 100 diopters than the lenses with planar membranes. It shows that aspherical membranes can improve the resolving power of liquid lenses at high diopters, thus providing a new route of optimizing the imaging performance of adaptive liquid lenses for various applications.
© 2016 Optical Society of America
1. Introduction
Focus-tunable liquid lenses have excited widespread attention courtesy of its additional degree of freedom to vary diopters (dpt) without sophisticated motorized cams that often require precise and synchronous displacement of several lens elements/groups along an extended trajectory [1–3]. Given its compact size, low cost, and fast and accurate focusing capability in a dynamic range, liquid lens has inspired next-generation miniaturized auto-focus and zoom lens design [4–7], and offered a great potential in laser projection and processing [8, 9], consumer and industrial illumination [4, 10], machine vision [11–13], ophthalmology [14–16], microscopy [17–19], etc.
The two mainstream approaches to implement focus-tunable liquid lenses are electrowetting [20] and shape-changing polymers [21]. An electrowetting lens consists of two liquids of similar densities yet different refractive indexes (RI) sandwiched between two glass windows. The curvature of the liquid-liquid interface, and thus its optical power can be manipulated by an external voltage. Due to the small capillary force along the meniscus, making a lens with the aperture (A) exceeding 3 mm is challenging. The basic optical layout of an elastomer-liquid lens comprises a fluid cavity with one side being a thin elastomer membrane and the other being a transparent window. Pressure change at different magnitudes in the cavity causes the membrane to deflect into varied heights, thus forming a vari-focal plano-convex lens. Such configuration is scalable in aperture, and can be readily integrated with a wide range of actuation mechanisms, such as mechanical motor [22], head load [23, 24], electromagnetic coil [25], piezoelectric actuator [26], and dielectric elastomer [27, 28].
In most elastomer-liquid lenses, the center deflection (h) of the lens membrane is much smaller than its aperture [3]. The small deflection helps suppress the optical aberrations [29], e.g. field curvature and spherical aberration. Studies on membrane mechanics reveal that the deflected profile of an edge-clamping circular membrane can be assumed as a spherical shape only when h << A [22]. At a large h, the undesirable parabolic deflection profile exhibits a significant deviation from the spherical shape due to edge-clamping [22, 30, 31]. This increases the optical aberrations and impairs the image quality [24, 29, 30, 32]. Membrane pretension can alleviate the boundary effect at small deflections. It however has limited contributions at large deflections [33], leaving alone the assembling difficulty caused by the pretension. An alternative approach to alleviate boundary effect is to abandon the marginal part of the lens [34]. For example, Optotune EL-10-30 uses only up to 80% of the lens diameter (10 mm) as the clear aperture at certain back focal length (BFL). The reduction of available clear aperture is however not a preferable solution for the lens that already has an original small diameter. Therefore, current elastomer-liquid lenses have limited uses in the applications that require small original lens diameter yet large diopters so as to reduce the depth of field or optical track length.
Another solution to address this is to use an elastomer membrane with spherical or conical profiles [35, 36], rather than a thin membrane with constant thickness (hereafter referred as CTLL, constant thickness liquid lens). Optical simulation shows that these profiles may help reduce aberration and enhance resolution. Such claim, unfortunately, remains vague without experimental validation. This is due to the difficulty in fabricating and controlling a spherical/conical elastomeric membrane with a desired thickness profile [36]. In this study, a circular membrane with aspherical cross-section was developed, where the membrane thickness of the lens (hereafter referred as VTLL, varied thickness liquid lens) varies from the edge to the center following an aspherical geometry. The membrane is replicated from the deflected lens profile of a CTLL at given center deflections. This highly controllable process allows easy fabrication of elastomer membranes with designed geometries. Optical testing using the fabricated VTLL shows that by alleviating the boundary effects at small and large membrane deflections, the VTLL exhibits improved center and peripheral resolutions than the CTLL at high diopters.
2. Materials and methods
2.1 Lens design
Figure 1 illustrates the cross-section of a 6mm VTLL. It consists of a cover glass (thickness: 1 mm) seated on the retention lip of the top mounting cell, an air gap (thickness: 2 mm), an aspherical membrane sandwiched between the top and bottom mounting cells, a liquid cavity (thickness: 4mm), and another cover glass (thickness: 1 mm) held in place by the bottom mounting cell. The air gap accommodates the upward deflection of the lens membrane, which, according to paraxial ray trace analysis, enables the BFL (expressed in dpt hereafter) to reach more than + 200 dpt. The annular notch on the bottom cell coincides with the rim of the top cell for centration and alignment of all optical elements. The optical fluid is between the lens membrane and the bottom cover glass. The lens has two luer fittings for connections with an external syringe pump. During operation, extra volume of optical fluid is pumped into the liquid cavity, which changes the curvature of the lens membrane and consequently the optical power.
Fig. 1 Schematic of the VTLL. (a) The perspective view of a VTLL with (b) a membrane having an aspherical cross-section. The top and bottom mounting cells fix the membrane in between by silicone adhesives. The optical fluid is concealed between the membrane and the bottom cover glass, which forms the elastomer-liquid lens when the membrane is deflected by hydraulic pressure. VTLL, varied thickness liquid lens.
The aspherical membrane of a VTLL was replicated from the deflection profile of a CTLL. According to the plate theory, the deflection profile of an edge-clamped thin membrane with constant thickness can be determined [30, 33] and approximated by aspherical fitting as:
where x is the membrane position and y is the upward membrane deflection; C is the membrane curvature; k is the conic constant which is assumed as −1 (paraboloid); A4, A6, A8, A10, and A12 are polynomial coefficients.In this study, a CTLL with 500 μm thick membrane was actuated to generate the aspherical membrane profile. The replicated membrane has a small thickness at the edge and a large thickness at the center. The aspherical profile of the replicated membrane can be expressed by the center thickness tC and the thickness ratio TR, which is defined as:
where tE is the edge thickness of the membrane.In order to determine the optimal aspherical membrane geometry that leads to the least field curvatures and thereby the best peripheral resolution at high diopters, spherical deviations of the VTLLs with various aspherical membranes (with tC = 0.3mm and varied TR values) were compared at a given center deflection (h = 0.805mm) (Figs. 2(a) and 2(b)). The lens profiles of VTLLs were first obtained by multiphysics finite element analysis software (COMSOL 4.3b, CA), where the Young’s modulus of the membrane material (polydimethylsiloxane (PDMS)) was set as 1.8 MPa, Poisson’s ratio as 0.49, and the density as 965 kg/m3 [37]. The profiles were then imported into optical design program OpticStudio 14.2 (Zemax, LLC, USA) for obtaining the RMS spot radii at different field angles from 0° to 6.0° with 532 nm incident collimated beam (Figs. 2(c) and 2(d)). The RI of the lens material was set as 1.408. The simulation results echo previous studies on the boundary effects [30, 32] and shows that the maximum spherical deviation occurs near the lens’ edge. The CTLL has the largest deviation (0.11 mm), which occurs at a position that is the closest to the lens center (with the distance of 2.38 mm). This leads to the worst RMS spot radius (138 μm) at the field angle of 6.0° and the largest variation across the entire field. The spherical deviation of the VTLLs decreases with increasing TR. However, when TR is greater than 1, the RMS spot radius exhibits substantial increase, and may exceed that of CTLL at small field angles. The VTLL with TR = 1 has the least increase of RMS spot radius over the entire field (23.3%) and its RMS spot radius is smaller than that of the CTLLs across the entire field. It was thus selected as the design of choice due to its least field curvature without considerable loss of center resolution.
Fig. 2 Optical performance comparison between the CTLL and the VTLL. (a) Simulated optical profiles of CTLL and VTLLs, and (b) their spherical deviations at the center deflection of 0.805 mm; (c) their corresponding RMS spot radii from 0° to 6° field angle and (d) changes of spot radii with respect to 0° field angle at + 100 dpt. The spherical profile (green dotted curve) at the same center deflection is superimposed in (a) for comparison. The thickness ratio (TR) in the legend is defined as (tC - tE)/tE. For both the CTLL and VTLLs, tC = 0.3 mm. 85% aperture is used for optical simulation. The CTLL has the largest spherical deviation (0.111 mm) and the largest RMS spot radius at 6° field angle (135.47 μm). Among VTLLs, the lens with TR = 1 has the smallest RMS spot radius (26.41 μm) at 6° field angle and the smallest increase of RMS spot radius from 0° to 6° (23.3%). CTLL, constant thickness liquid lens; VTLL, varied thickness liquid lens; tC, center thickness; tE, edge thickness; dpt, diopter.
For the VTLL with TR = 1, the spherical deviations at various center deflections were determined (Figs. 3(a) and 3(b)). As expected, the deflected aspherical membrane approximates a spherical shape at small center deflections. The spherical deviation increases with increasing center deflection [30, 33]. Ray tracing analysis was performed to estimate the RMS spot radii of the VTLL and an N-BK7 plano-convex lens at various focusing powers. It is interesting to find that the RMS spot radius of the VTLL does not exhibit a monotonous change with the focusing power. Instead, the smallest RMS spot radius occurs at + 40 dpt, and remains below 25 μm for most field angles when the BFL is below + 100 dpt (Fig. 3(c)). This dioptric range from + 40dpt to + 100dpt was selected for subsequent experimental characterization. It should also be noted that although the RMS spot radius of the VTLL is significantly lower than that of a CTLL, it is still larger than the solid lens at corresponding BFLs (Fig. 3(d)).
Fig. 3 Optical performance of a VTLL with TR = 1. (a) Simulated optical profiles of the VTLL at the center deflection of 0.136, 0.199, 0.376, 0.585, 0.676, and 0.805 mm; (b) the corresponding spherical deviations; (c, d) the RMS spot radii for the VTLL and for a spherical plano-convex N-BK7 lens at + 16.7, 25, 40, 80, 100 and 117.6 dpt. VTLL, varied thickness liquid lens; TR, thickness ratio; dpt, diopter.
2.2 Fabrication of aspherical membrane
The aspherical membrane of the VTLL was replicated from the surface profile of a CTLL, using a previously reported fabrication process [38]. A CTLL with 6 mm in diameter was first fabricated. A SU-8 circular microstructure with 200 μm in thickness and 6 mm in diameter was patterned on a silicon wafer. The wafer was placed face to face with a microscopic glass slide coated with 1.5 μm thick S1813 photoresist (Shipley, MA), where the gap in between was determined by a metal wire that was 700 μm in diameter. The gap was later filled by PDMS (Sylgard 184, Down Corning, MI) prepolymer with the mixing ratio of 10:1 w/w. After crosslinking, the PDMS membrane above the circular microstructure had a thickness of 500 μm. The PDMS substrate was then peeled from the glass slide, aligned and bonded with another PDMS substrate that had pre-patterned microfluidic channels (200 μm in thickness and 500 μm in width) to form the CTLL with a 500μm membrane. Finally, the as-fabricated CTLL was filled by 99% glycerol (Sigma-Aldrich, MO) using a syringe pump (Pump 11 Elite, Harvard Apparatus, MA) at a flow rate of 20 ml/h.
The deflected surface profile of the CTLL was replicated by two-step replica molding (Fig. 4(a)). First, a 200 μm thick PDMS stencil with a 6 mm through hole was bonded on the top surface of the CTLL, where the hole was concentric to the lens membrane. The CTLL was actuated to reach a center deflection of 150 μm. A glass slide was covered on the PDMS stencil with 1 mm thick spacers. PDMS prepolymer was then filled into the gap. After crosslinking, a 200 μm pillar with a 150 μm concave depression atop was created on the replicated PDMS substrate. The microstructure was soaked in 0.5% (hydroxypropyl)methyl cellulose (Sigma-Aldrich, MO) for 10 min, rinsed with de-ionized water and air-dried. Afterwards, the concave PDMS structure was placed face to face with a glass slide with a 350 μm thick spacers. The gap was again filled with PDMS polymer. After crosslinking, the PDMS membrane with a convex aspherical bottom surface was achieved with the center thickness of 300 μm and the edge thickness of 150 μm (Fig. 4(b)). The optical measurements below were based on the VTLL with such an aspherical membrane.
Fig. 4 Fabrication process. (a) A two-setup replica molding process for creating the lens membrane with a desired aspherical cross-section; (b) the optical micrograph showing the cross-section of the aspherical membrane; (c) a snapshot of an assembled VTLL and (d) the VTLL without lensing effect. The refraction at the aspherical interface is minimized due to the refractive index match between the membrane and the optical fluid. Leaf veins outside and inside VTLL share the same focus. VTLL, varied thickness liquid lens.
2.3 Lens assembly and filling
The top and bottom mounting cells of the VTLL were made of a 3D printed optically-opaque resin (Formlabs 1, MA). Two cover glasses were affixed to the mounting cells by UV adhesive. The lens membrane was bonded to the bottom cell using silicone sealant with the convex aspherical surface facing towards the bottom cell. An assembled VTLL is shown in Fig. 4(c). Using the similar approach, a CTLL with a membrane thickness of 150 μm was also assembled.
The lenses were filled by 59% wt. glycerol (Sigma-Aldrich, MO)-water mixture with the RI of 1.41, which closely matches with that of PDMS (RI = 1.408). The refraction at the liquid-elastomer interface was minimized, and thus the lens can be approximated as a plano-convex singlet during membrane deflection. Figure 4(d) shows a VTLL without the lensing effect, where the leaf veins acquired with and without the VTLL shared the same focus when the lens membrane was not deflected.
3. Results
3.1 Back focal length (BFL)
The BFLs of horizontally placed CTLL and VTLL (optical axis perpendicular to the gravitational direction) were measured at the hydrostatic pressure from 20 to 240 mbar at a 20 mbar increment using a custom-built Foucault knife edge test setup. The lens was illuminated by a 532 nm collimated laser beam after being magnified by a 3 × Keplerian beam expander (beam diameter: 10.5 mm). A razor blade mounted on a two-axis translation stage was adjusted to intercept the converging ray bundles from the left side of the optical axis, and placed upstream of the screen to visualize the ray interception pattern. The BFL can be thus approximated as the reciprocal of the distance between the bottom glass windows of the lens and the razor blade when it obscured the left and right halves of ray bundles simultaneously. For the CTLL, the BFL was from + 7.4 dpt at 20 mbar to + 56.9 dpt at 120 mbar and to + 108 dpt at 200 mbar (during inflation). Because the VTLL had a smaller edge thickness than the CTLL, it required less hydrostatic pressure to reach the same power, i.e. + 15.6 dpt at 20 mbar to + 127.9 dpt at 120 mbar (Fig. 5). The results also indicate that the lenses exhibit fairly small hysteresis between inflation and deflation.
Fig. 5 BFLs (in dpt) of CTLL and VTLL as a function of hydrostatic pressure. The BFL was the mean value of four measurements (n = 4). BFL, back focal length; CTLL, constant thickness liquid lens; VTLL, varied thickness liquid lens; dpt, diopter.
3.2 Image contrast and peripheral resolution
The off-axis resolution was examined by having the lens of interest image a Siemens-star resolution target at + 100 dpt and + 40 dpt with the full aperture on a 4.1 Mega-pixel CMOS sensor (CMOSIS CMV4000-3E5, Point Grey Research Inc., Canada). The target was centrally aligned with the optical axis of the lens using a 3-axis translation stage and illuminated by a telecentric illumination system where the warm white light incident on a diffuser (200-grit) was collimated by an aspherical condenser. At a given power, the object and image distances were adjusted to set the magnification at about 5 × in order to make the target fill 90% of the frame and in best focus.
Figures 6(a)-6(c) show the images taken by the CTLL, the VTLL and an off-the-shelf 6 mm N-BK7 plano-convex solid lens at + 100 dpt, respectively. The Michelson contrast (CM) was used to calculate the overall contrast of the star image under the same lighting condition, which is defined as:
where Imax and Imin are the maximal and minimal luminance among all pixels. The mean contrasts in the CTLL, the VTLL and the solid lens were calculated as 0.62, 0.69 and 0.75. As seen from the insets, CTLL displayed the most evident gradual focus softening at the edges. VTLL presented crisper edges than CTLL, yet was outperformed by the solid lens. CTLL and VTLL showed slight astigmatism due to fabrication imperfection that caused unexpected membrane asymmetricity. The edge sharpness in the peripheral areas was qualified by examining the luminance profile along top and right edges of the wedged lines (6 line pair (lp)/mm) on the meridional and sagittal planes using MATLAB image analysis (Figs. 6(d) and 6(e)). These edge areas have the corresponding field angle of 6.0°. At both distal ends, the solid lens exhibited the sharpest transition at the edges, the CTLL exhibited the smoothest transition, and the VTLL sit in between. The rise distance (D) was used as a numerical indicator to qualify the edge sharpness, which counts the pixel number within which the luminance magnitudes changes from 10% to 90% within the transition range, i.e. from (0.1Imax + 0.9 Imin) to (0.9Imax + 0.1Imin) [39]. The calculated means of rising distance D (n = 4) for the CTLL, the VTLL and the solid lens were 30, 23, and 21, respectively. This indicates that the VTLL had a much better resolving power at the edge than the CTLL, but was still outperformed by the solid lens.Fig. 6 Center and peripheral resolution comparison at + 100 dpt. The lenses focus on a Siemens star target at 5.0 × magnification and + 100 dpt. (a) CTLL, (b) VTLL, and (c) N-BK 7 plano-convex spherical lens. The first two columns show the original snapshots and their inverted images for visualization purposes (scale bar: 2 mm). The rest three columns show the center and peripheral regions on the meridional and sagittal planes of the images from the second column (scale bar: 0.5 mm); (d, e) relative luminance along the top and right edges (marked by dotted colored line) of the Siemens star target. CTLL, constant thickness liquid lens; VTLL, varied thickness liquid lens; dpt, diopter.
The off-axis resolutions of the CTLL, the VTLL and the solid lens at + 40 dpt were also examined using the same imaging setup except that the object and image distances were elongated to set the magnification at 5.1 × while maintaining the focus of the target (Fig. 7). The mean contrasts in the CTLL, the VTLL and the solid lens now increased to 0.68, 0.76, and 0.79, respectively. The overall image contrast presented by the VTLL was very close to that of the solid lens, and was again much better than the CTLL. The calculated means of rising distance D (n = 4) for the CTLL, the VTLL and the solid lens decreased to 22, 17, and 16, respectively. All the lenses depicted increased edge sharpness at + 40 dpt than those at + 100 dpt, while the ranking order of lens performances remained the same, as at + 100 dpt.
Fig. 7 Center and peripheral resolution comparison at + 40 dpt. The lenses focus on a Siemens star target at 5.1 × magnification and + 40 dpt. (a) CTLL, (b) VTLL, and (c) N-BK 7 plano-convex spherical lens. The first two columns show the original snapshots and their inverted images for visualization purposes (scale bar: 2.1 mm). The rest three columns show the center and peripheral regions on the meridional and sagittal planes of the images from the second column (scale bar: 0.52 mm); (d, e) relative luminance along the top and right edges (marked by dotted colored line) of the Siemens star target. CTLL, constant thickness liquid lens; VTLL, varied thickness liquid lens; dpt, diopter.
The superior peripheral resolution of the VTLL over the CTLL agrees well with theoretical analysis. This is believed due to the fact that the surface profile of the VTLL born more resemblance to the spherical contour than the CTLL, which is instrumental to correct the off-axis aberration. The improved contrast and edge sharpness at + 40 dpt mainly result from the increased F/# and reduced amount of soft focus caused by spherical deviation (i.e. the lens at smaller powers has a smaller spherical deviation than that at larger powers).
3.3 Central resolution and modulus transfer function (MTF)
Figures 6 and 7 also indicate that in addition to better peripheral resolution, the VTLL had better central resolution than the CTLL, although both were outperformed by the solid lens. To quantify their central resolutions, a bright field USAF 1951 resolution target was used for measurement at + 100 dpt (Figs. 8(a)-8(c)). With proper alignment, the region containing groups 6&7 of the resolution target were projected to the very center of the field. All lenses were able to resolve the elements in groups 4&5, whereas the CTLL exhibited the worst contrast of the bar pattern. The line resolving ability can be interpreted from the inset that contains the elements with higher frequencies in groups 6&7. The CTLL resolved the vertical bars from element 2 in group 6 (frequency: 72 lp/mm) at the contrast of about 0.21, as compared to the VTLL at the contrast of about 0.30, and the solid lens at the contrast of 0.43. The MTF performance comparison of the CTLL and the VTLL (Fig. 8(d)) verifies that the VTLL has a higher resolving power. For example, with the minimal contrast of 0.3 the VTLL can resolve 56 lp/mm, while the CTLL can only resolve 42 lp/mm.
Fig. 8 Central resolution measurement by imaging a positive USAF 1951 resolution target. (a, b, c) show the measurements via the CTLL, the VTLL, and a solid lens at + 100 dpt, respectively. The insets are the magnified views (digital zoom: 2.5 × ) of the highlighted region of the left images; and (d) comparison of meridional MTF curves of CTLL and VTLL at + 100 dpt. The MTFs correspond to the center of the frame. VTLL presents a better contrast from 16 lp/mm to 91 lp/mm. MTF, modulus transfer function; CTLL, constant thickness liquid lens; VTLL, varied thickness liquid lens; dpt, diopter.
4. Discussion
The improved resolving power allows the use of the full aperture of an elastomer-liquid lens with a small lens diameter. The VTLL used here is 6mm in diameter, which is smaller than most commercial products or research prototypes that can generate satisfactory image quality. This is significant for space constraint applications where the use of partial lens diameter of a large lens as the clear aperture is not affordable. The results clearly show the improvement of optical resolving power of the VTLL over the CTLL at + 100 dpt, which promises a great potential of using elastomer-liquid lenses at high diopters, thus adding applicability and versatility of liquid lenses.
This study provides a simple and low-cost approach to create a lens membrane with desired inhomogeneous thickness profile for improving optical performance. The aspherical membrane shape of the VTLL was achieved by deforming a CTLL. Different from many previously reported elastomer-liquid lenses, pre-straining is not necessary. Fabrication and assembly difficulties are thus reduced. It is noted that the deflected membrane profile approximated by Eq. (1) may not be the optimal shape for reducing field curvature. Numerical analysis showed that other geometric profiles, such as spherical or conical contours [40, 41], may also increase the resolving power to certain extents, despite of challenging fabrication of optical grade elastomer membranes with such geometries. To identify the best geometric profile of the membrane that can lead to least field curvature and most enhanced resolving power, one can backtrace a deflected membrane profile of the VTLL that results in the best set of the target optical parameters (including RMS spot radius, wavefront error, and MTF across the required field at targeted dpts) to the thickness profile of a freeform membrane before deformation in an iteratively manner. The final thickness profile may be determined by considering the fabrication complexity and the need for pre-straining.
Since this study makes the surface profile of focus-tunable lens approximates the spherical shape at a large power, spherical aberration is also expected. The membrane geometry of the VTLL presented here may not be the best optimum for spherical aberration correction. Fortunately, existing ways to correcting spherical aberrations of solid lenses can be readily applied to liquid lenses [34].
5. Conclusion
We introduce a novel focus-tunable elastomer-liquid lens design that takes advantage of a microfabricated aspherical lens membrane. The mechanical and optical analysis show that compared to the flat membrane with constant thickness, the aspherical membrane effectively reduces spherical deviations at both small and large membrane deflections, and leads to smaller RMS spot radii across the field. Experiments demonstrate that the VTLL has better off-axis edge sharpness and better center MTF performance than the CTLL at large optical powers. Due to the enhanced resolving power in the central and peripheral regions, the VTLL design allows for the use of a large fraction of the total lens diameter, which promises a potential for future miniaturization of elastomer-liquid lenses without sacrifice of imaging quality.
Acknowledgment
This work is funded by NSF grants under the award numbers 0954013 and 1138236. The authors also thank the HHMI Med into Grad program, OSU Alumni Grants for Graduate Research and Scholarship, and Pelotonia program for student fellowship supports.
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