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A miniature solid-state varifocal lens based on Alvarez principle with lens elements having free-form surfaces is reported. The Alvarez lens elements are implemented with diamond-turning and replication molding processes. They are integrated with electrostatically-driven MEMS comb-drive actuators fabricated using SOI micromachining. Dynamic tuning of focal length more than 1.5 times (from 3 mm to 4.65 mm) is experimentally demonstrated with only small MEMS-driven lateral movements of 40 μm. Such varifocal lens may be useful in miniature cameras for autofocus and zooming due to its advantages including ease of packaging and fast tuning speed.
©2013 Optical Society of America
1. Introduction
Miniature tunable lenses are attractive due to their ever-increasing number of applications in a range of areas including smart phones, surveillance, robotics, and biomedical systems. Currently, two main technical approaches are adopted by the researchers from both academia and industry to implement miniature tunable lenses. One approach alters the lens material’s refractive index or index distribution [1, 2] and hence its focal length. The other, inspired by varifocal crystalline lenses found in animal eyes, changes the shape of the lens [3, 4]. Examples of the latter include micromachined liquid lenses tuned by pressure [5–7], electrowetting [8, 9], and many other effects [10].
A few decades ago, Alvarez [11] and Lohmann [12] independently proposed an interesting type of lenses with varifocal capability. These lenses, known generally as Alvarez lenses, are paired optical elements having complementary cubic surface profiles. They provide variations in optical power through small lateral displacements, one with respective to the other, perpendicular to the optical axis. However, despite its simple operating principle, Alvarez lens largely remained impractical for a long time since its invention, which is probably due to the demanding manufacturing issues in precise implementation of the required cubic surface profiles. In recent years, advances in manufacturing technology have enabled the production of high-quality low-cost free-form optical surfaces, thus largely overcoming the practical restrictions in implementing this type of lenses. There is hence a renewed interest in Alvarez lenses [13, 14], both in refractive [15, 16] and diffractive [17] forms.
In this paper, we take a step further by merging it with microelectromechanical systems (MEMS) technology [18] and demonstrate a miniaturized MEMS-driven tunable Alvarez lens. Compared with other solid varifocal lenses that are tuned through mechanically deforming the lens shapes, the proposed lens has the potential to achieve a larger focal length tuning range with a smaller driving force. When compared with tunable lenses that are based on liquid platforms, the proposed lens has no liquid involved in the system, hence resulting in ease of packaging and handling with no potential leakage and evaporation issues and thereby making it more mechanically and thermally robust. In addition, focusing and tuning of focal lengths can be fast owing to the use of MEMS actuators.
2. Lens design
According to Alvarez [11], the variable lens consists of two identical special lens elements arranged in tandem one immediately behind the other. As shown schematically in Fig. 1(a) , the first and second lens elements have thicknesses (measured along the optical axis z) described respectively by the following equations:
where A, D, and E are constants, and x and y are transverse coordinates normal to z. Here, we assume A is a positive constant. Clearly, the combined thickness of the two-element system is then t = t1 + t2 = 2E, which is equivalent to a parallel plate. It can been shown that when the first element moves a displacement δ and the second moves -δ along the x direction, the combined thickness t has a parabolic term −2Aδ(x2 + y2) thus emulating a converging lens for positive displacement δ and a diverging lens for negative δ. The focal length of the combination hence can be calculated with the following equation [11]:where n is the refractive index of the lens material. From the above equation, it can be seen that the constants D and E in Eq. (1) have no direct effect on the focal length. However, as pointed out by Alvarez in [11], the constant D defines the tilt of the free-form surface and can be used to reduce the overall thickness of the lens element.
Fig. 1 (a) An Alvarez lens consists of two lens elements having free-form surfaces. (b) Schematic showing a MEMS-driven tunable Alvarez lens. (c) and (d) Free-form surface z(x,y) = 0.6(xy2 + x3/3)-0.2x + 0.35 viewed from different viewing angles.
Clearly, such a lens has the ability to vary its focal length substantially with only a small relative lateral shift between its two constituent elements, and can be conveniently miniaturized and integrated with MEMS technology for tuning. As shown schematically in Fig. 1(b), the two lens elements are mounted respectively on two MEMS electrostatic comb-drive actuators [19] fabricated using silicon-on-insulator (SOI) micromachining, and subsequently assembled to form a MEMS-driven tunable Alvarez lens. Thus, the relative lateral movements of the lens elements induced by the MEMS comb-drive actuators, as indicated in the figure, control and tune the focal length of the varifocal Alvarez lens.
In our design here, each Alvarez lens element has a flat surface and a free-form surface defined by the right-hand side of Eq. (1) with coefficients A = 0.6 mm−2, D = −0.2, and E = 0.35 mm, i.e. z(x,y) = 0.6(xy2 + x3/3)-0.2x + 0.35. Three-dimensional views of the free-form surface seen from two different viewing angles are illustrated respectively in Fig. 1(c) and 1(d). The profile of the free-form surface is calculated and plotted within a circle with a radius of 1 mm centered at the origin (x = 0, y = 0). With these two lens elements, and assuming each MEMS actuator can move bi-directionally with a maximum stroke of 40 μm and the refractive index of the lens material is n = 1.56, we can expect directly from Eq. (3) that such a varifocal lens is able to tune from a diverging lens to a converging lens with a focal length tuning range from −18.60 mm to -∞ and from + ∞ to18.60 mm. As we have mentioned before, one of the potential applications of the proposed varifocal lens is for zooming and autofocusing in miniature cameras. For such application, the required focal length is typically only a few millimeters. One way to achieve focal length tuning in this range with the same Alvarez lens elements is to offset the centers of the two elements laterally along the desired x-axis of the free-form surface to set the varifocal Alvarez lens to a short initial focal length. For example a center-to-center offset of 300 μm (i.e. δ = 150 μm) results in an initial focal length of 4.96 mm. With the same ± 40 μm strokes offered by the MEMS actuators, we can expect a focal length tuning range from 3.92 mm to 6.76 mm. It should be noted that a larger focal length tuning range can be achieved simply through enhancing the maximum travel range of the MEMS actuators.
3. Fabrication
The practical implementation of the above-mentioned varifocal lens design is possible through two ways. In one way, the Alvarez lens element is also fabricated through an IC-like microfabrication process, for example through gray-scale lithography and/or etching processes to produce continuous surface profiles or through multiple lithography-etch steps to produce stair-case surface profiles emulating the freeform surfaces. The concept of diffractive optics [20] may be used here to achieve large phase variations across the aperture. In this case, the whole MEMS-driven optical device having an integrated Alvarez lens element can be fabricated through a single IC-like microfabrication process without the necessity for alignment and assembly of components. However, the optical performance of such varifocal lens might be compromised. In the other way, the MEMS actuator and Alvarez lens element are fabricated separately with different methods, and then finally assembled to construct the MEMS-driven optical device shown in Fig. 1(b). This latter approach is more practical as it offers a better optical quality of the lens element whilst maintaining a low-cost and high-volume production process. In this paper, we take the latter approach.
The comb-drive actuator together with the rigid platform with a through-hole opening is fabricated using a commercially available SOI MUMPS process [21]. A brief illustration of the process is shown schematically in Fig. 2(a) . The fabrication begins with a double-side polished SOI wafer. A 1st lithography step followed by a Deep-RIE step is used to fabricate the desired microstructures (including comb electrodes, rigid platform, and fold-beam flexure suspensions) into the top Si device layer. Subsequently, a 2nd lithography and Deep-RIE steps are performed on the backside of the SOI wafer, namely substrate side, to make an opening in the substrate right under the movable microstructure region. By removing exposed buried oxide, all the movable microstructures can be released and the fabrication of the MEMS actuator and the platform for mounting the lens element is done. An overview of the fabricated MEMS device together with close-up views showing the details of a comb-drive and a folded-beam suspension is provided in Fig. 3(a) . In our design, the folded-beam suspensions have flexure beams which are 1300 μm long and 7.5 μm wide, and the comb-drives have interdigitated fingers with finger length of 70 μm, width of 5 μm, finger gap of 4 μm, and initial finger overlap of 5 μm. The actuator has a total number of 644 movable fingers for one-side actuation. All movable structures including the comb-drives and the platform have a uniform thickness of 25 μm.
Fig. 2 Schematics showing (a) MEMS actuator fabrication and (b) Alvarez lens element fabrication and assembly.
Fig. 3 (a) SOI micromachined electrostatic comb-drive actuator. (b) Diamond-turned master mold having the desired free-form surface. (c) Replicated transparent lens element. (d) MEMS-driven Alvarez lens element after finally assembly process.
The transparent Alvarez lens element is obtained through a diamond-turning and replication molding process [22]. First, the single-point diamond turning technique [23] with free-form surface fabrication capability is used to make the desired surface contour on an Aluminum substrate as shown in step (i) of Fig. 2(b). This acts as the master mold. Slow slide servo turning with a single crystalline diamond turning tool (nose radius of 0.2 mm, rake angle of 0 degrees and clearance angle of 7 degrees) is used to cut the free-form surface. Then, a PDMS mold with an inverse pattern is fabricated using a standard replication process shown in step (ii), in which liquid polydimethylsiloxane (PDMS) prepolymer (Sylgard 184 silicone elastomer, a base and curing agent of Dow Corning Corp, mixed in a 10:1 weight ratio) is poured onto the Aluminum mold followed by complete curing under 65°C for 2 hours and detachment. The final lens element is achieved using another replication process. Different from the previous one, in this step, the concave part of the PDMS mold is filled with a UV curable optical adhesive (NOA83H, Norland, USA) and sealed by another flat PDMS slab (as shown in step (iii)). After exposure under UV light, the whole setup is put into a convection oven (60°C, 30mins) for further hardening of the lens element. Thanks to the low surface energy of the PDMS, the adhesion between the lens element and the PDMS mold is extremely small, making the detachment very easy without affecting the element surface quality. As shown in step (iv) in Fig. 2(b), the as-fabricated solid lens element is finally integrated onto the MEMS actuator platform via a microassembly process under the assistance of a microscope for alignment and an epoxy adhesive for fixing.
The fabricated device is shown in Fig. 3, where (b), (c), and (d) are images of the diamond-turned master mold, replicated transparent lens element, and the MEMS-driven Alvarez lens element after the final assembly process, respectively. It is noted that for Alvarez-type varifocal lenses, the movement direction of the actuator must be aligned to the desired axis of the free-form surface, for example the x-axis for the surface described in Eq. (1). Hence, to facilitate the alignment of the lens element to the MEMS microactuator during assembly, alignment features indicating the x- and/or y-axis of the free-form surface is incorporated in the lens element during its master mold fabrication process using diamond turning. For the circular element used here, an additional cut at its circumference is made to indicate the direction of the y axis of the free-form surface (see Fig. 3(b)).
In our manual assembly process, the lens element is placed on a rotational stage with free-form surface facing upwards. The SOI micromachined MEMS comb-drive actuator, with silicon device layer facing downwards, is held by a gripper attached to a three-axis precision translational stage. The alignment features on the lens element and MEMS-driven platform are aligned under an optical microscope before the element is secured on the platform by applying and curing an epoxy adhesive.
4. Lens testing and characterization
The assembled MEMS devices are calibrated under an optical microscope. Static displacement of the lens platform as a function of DC driving voltage is first characterized. The platform displacement is measured when the electrodes corresponding to the desired direction are applied with a driving voltage and those corresponding to the opposite direction are grounded. Experimental results showing the displacement vs. voltage relationships of a fabricated MEMS-driven Alvarez lens element are given in Fig. 4(a) . Clearly, the MEMS actuator is able to provide a bi-directional motion of up to ± 42 μm under a driving voltage of 55 Volts. Next, the dynamic characteristics of the devices are also tested under the microscope. The comb-drive microactuator is driven to oscillate using a “push-pull” mechanism [19] with 2 Volts DC bias voltage and 4 Volts AC peak-to-peak voltage. Figure 4(b) shows the measured frequency response of the MEMS-driven Alvarez element. The resonance of the device is found to be at 104 Hz, which indicates that the proposed MEMS tunable Alvarez lens can potentially have a fast tuning speed in the millisecond range.
Fig. 4 (a) Measured static displacement as a function of DC driving voltage for a MEMS electrostatic comb-drive actuator used in our tunable Alvarez lens. (b) Its frequency response.
Next, two calibrated MEMS-driven Alvarez lens elements are arranged in tandem with free-form surfaces facing each other. A side view of the assembled Alvarez lens is shown in the inset of Fig. 5(b) . The centers of the two lens elements are intentionally offset laterally along the desired x-axis of the free-form surface to set the Alvarez lens to an initial focal length before testing. As shown in Fig. 5(a), an optical system is set up to measure the focal length of the miniature tunable Alvarez lens, where an object (the National University of Singapore Logo) is placed at a known fixed object distance D and imaged by the lens under test. The size of the image produced by the Alvarez lens is then measured by a calibrated optical microscope to obtain the optical magnification M. The focal length of the Alvarez lens can then be calculated with the following equation:
It is noted that the optical magnification M is negative for a positive lens and positive for a negative lens since the object is located more than two times of the focal length away from the Alvarez lens in our setup.Fig. 5 (a) Experimental setup for focal length measurement. (b) Focal length of the MEMS tunable Alvarez lens as a function of the offset between two constituent lens elements. (c) to (h) Captured images at various focal lengths as indicated in (b). A series of images from P1 to P2 are also combined to create a video showing an apparent “zoom-out” effect due to the focal length change of the MEMS-driven Alvarez lens (Media1).
Four sets of experiment data are recorded at four different initial center-to-center offsets between the two lens elements. At each initial offset, the two calibrated MEMS comb-drive actuators are driven laterally in opposite directions with equal amount of displacement outputs to vary the center-to-center offset of the two lens elements. To be more specific, one actuator is driven to move from −40 μm to 40 μm with the other to move from 40 μm to −40 μm along the desired direction perpendicular to the optical axis, both with steps of 10 μm. Such an operation varies the focal length of the Alvarez lens without laterally shifting the optical axis of the lens. The open symbols in Fig. 5(b) show experimental results. Images captured through the Alvarez lens at different focal lengths, as indicated in Fig. 5(b) with P0 to P2 and N0 to N2 respectively for positive and negative lens configurations, are given in Fig. 5 from (c) to (h). For comparison, the theoretical estimations using Eq. (3) with the refractive index of the lens material set at its nominal value of 1.56 are also provided. They are shown as solid symbols in the figure. It can be seen that in general the experimental results agree with the theoretical predictions. The MEMS tunable Alvarez lens can adjust its focal length substantially with only small lateral displacements offered by the electrostatic comb-drive microactuators. However, it is also observed from the figure that there exist discrepancies between measured and estimated focal lengths and some images captured show relatively poor image quality. These might be due to the following reasons. Firstly, the operation principle of the Alvarez lens is based on the assumptions that each constituent lens element acts as a thin pure optical phase transformer and they have a negligible separation along the optical axis. For our practical implementation here, such assumptions might not be fully fulfilled, for example our lens element is relatively thick and the separation gap between the two elements is estimated to be around 350 μm from the side view of the lens assembly given in Fig. 5(b), which might result in the differences between the theoretical and experimental results as well as introduce optical aberrations in the imaging system. And secondly, there might be alignment errors in our lens assembly due to the limitations of our manual assembly process, for example the free-form surface’s x-axis might not be perfectly aligned to the actuator’s movement direction, which in turn may introduce additional aberrations in the system and result in poor image quality at certain focal lengths. Further enhancement of the lens performance may be possible through improving the manufacturing and assembly processes for a better imaging quality and enhancing the actuator’s travel range for a larger focal length tuning range.
5. Conclusions
In this paper, we demonstrated a non-liquid MEMS-integrated tunable lens based on the Alvarez principle. It is shown that such a lens, comprising two MEMS-driven lens elements having free-form surfaces, is able to vary its focal length substantially through a small relative displacement of the two lens elements along a direction perpendicular to the optical axis. Compared with other non-liquid tunable lenses, for example bio-mimic shape-varying polymer lenses tuned by mechanical force or pressure, the proposed lens potentially offers a larger focal length tuning range and a smaller force requirement as the two non-contacting lens elements are freely suspended by flexures. In contrast with liquid tunable lenses, the proposed lens is easy to package and handle – being more robust mechanically and stable thermally, does not have any issues related to liquid leakage and evaporation, and has a faster response time. Such MEMS-based varifocal lens may be useful in miniature cameras for autofocus and zooming.
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