In this paper, the dynamic mechanical stability of the liquid-filled lenses was studied, in which acoustic excitation was used as broad band perturbation sources and the resultant response of the lens was characterized using non-contact laser Doppler vibrometer. To the best of our knowledge, it’s the first time that the mechanical stability of liquid-filled lenses was experimentally reported. Both experimental results and theoretical analysis demonstrate that the resonance of the lens will shift to higher frequency while the vibration velocity as well as its magnitude will be reduced accordingly when the pressure in the lens cavity is increased to shorten the focal length. All of these results will provide useful references to help researchers design their own liquid-filled lenses for various applications.
©2012 Optical Society of America
Active optics is an important branch in optics society. Different from traditional passive configurations, the performance of optical systems can be actively adjusted to adapt to the variation of surrounding environment such as temperature, mechanical force and air turbulence. Active optics can also intentionally achieve particular functions including light scanning, auto focusing, image zooming and aberration compensation using tunable reflective/refractive optical components . Due to its high adaptability and flexibility, in recent years, active optics has been continuously attracting much interest and achieving considerable progresses. Till now, different types of reflective and refractive optical components have been successfully developed for constructing active optical systems. For example, deformable mirror with continuous facet, whose surface contour can be changed, has been widely used in astronomy observation , ophthalmology diagnosis [3–5] and high power laser [6, 7]. Refractive lens with tunable optical power capability is another important component [8–10]. Compared with its traditional counterparts, refractive lens can achieve functions like auto focus and optical zooming with more compact configuration, which is especially attractive to the rapidly developing field of miniaturized optical systems.
Among the widely reported results about tunable lenses, the membrane based liquid-filled lens is one of the most commonly adopted configurations [11–13]. It consists of a lens cavity sealed by an elastic membrane and a microchannel is used to connect this cavity to external environment, with which particular liquids (acting as lens medium) can be delivered into the cavity. With the continuous injection of the liquid, hydraulic pressure in the cavity will be increased, thus deforming the membrane into a quasi-spherical shape. Considering the fact that the membrane consists of one facet of the lens, the variation of its surface contour will change the final optical power of the lens. Owing to the excellent elasticity of membranes, the associated large deformation capability can be used to achieve a wide focal length tuning range. At the same time, considering other distinct advantages, such as simple device structure, easy fabrication and operation as well as low unit cost, it has attracted extensive researches.
Until now, previous researches have mainly focused on designing novel lens structures , integrating different actuating strategies , improving optical performance (reducing spherical and chromatic aberration) [16, 17] and exploring new applications . However, various environmental factors such as temperature fluctuation, wind load, mechanical vibrations and acoustic wave radiations can cause lens to distort and thereby affecting its performance. The dynamic mechanical stability of the lens itself during operation is thus one of the most important considerations in real applications, which has not been thoroughly investigated.
In this paper, the vibration of the liquid-filled lens especially its membrane under external acoustic perturbation is studied. During the experiment, the vibration responses of the lens working at different focal lengths with respect to external acoustic perturbation have been successfully characterized using laser Doppler vibration measuring method. From the experimental results, it can be seen that the natural resonant frequency of the lens shifts to higher values when the lens is adjusted to work under shorter focal length, and at the same time, reduced vibration velocity as well as its amplitude can be observed. In addition, the lens responses to different disturbing levels have also been measured.
2. Device fabrication and vibration testing setup
The membrane based liquid-filled lens is fabricated using Polydimethylsiloxane (PDMS) based soft lithography process. Figure 1 shows a schematic of the fabrication process flow. First, the inverted structures of microchannel and lens cavity were patterned into an SU-8 layer (Fig. 1(a)) using standard photolithography steps, including spin-coating, soft/hard baking, exposure and developing. This SU-8 structure was then used as the master mold for the following replication molding (Fig. 1(b)), in which liquid PDMS prepolymer (a mixture of base and curing agent with weight ratio of 10:1, Dow Corning, USA) was directly poured onto the mold. After curing under 65° C for 2 hours, this PDMS slab with desired pattern having been transferred was peeled off from the mold (Fig. 1(c)). To fabricate the lens membrane, the same liquid PDMS prepolymer was spin-coated onto a blank silicon wafer followed by thermal curing (Fig. 1(d)). Through controlling the spinning speed, different membrane thicknesses can be achieved. The final device is obtained by irreversibly bonding these two PDMS parts together under the assistance of oxygen plasma surface treatment (Fig. 1(e)). The as-fabricated liquid-filled lens is shown Fig. 1(f) and 1(g) and the designed parameters of the lens are summarized in Table 1 .
In real applications, there are many potential disturbing sources existing in the surrounding environment, including temperature fluctuation, wind load, mechanical shake and acoustic wave radiation, which can cause lens vibration and thus affect its performance and lead to stability issues. In our experiment, a measurement setup as shown in Fig. 2 was used to characterize the dynamic stability of the as-fabricated liquid-filled lens, in which the acoustic wave generated by a speaker was used as the perturbation because of its wide frequency coverage and easy operation. The speaker (4227, Brüel&Kjær, Denmark) was connected to a signal controller, with which different frequencies and strengths of the acoustic wave can be generated. The as-fabricated liquid-filled lens was fixed at a certain distance above the speaker with its membrane facing the speaker. A syringe pumping system was connected to the inlet of the lens to control its intracavity pressure, the value of which can be directly measured via a pressure gauge connected to the outlet. The liquid filled lens was driven to vibrate under the excitation of the acoustic wave from the speaker, and the corresponding frequency and magnitude of the vibration was characterized using a commercial laser Doppler vibrometer (MSA-500, Polytec, USA) with its detection laser beam being transmitted through the whole lens structure and then focused on the membrane surface.
During the test, the pressure in the lens cavity was gradually increased by continuously introducing water via a syringe pumping system, resulting in decreased focal length. For each measurement points, the pressure value was first recorded by the pressure gauge. Then the speaker was activated with a sweep frequency source and the resultant vibration of the lens membrane was characterized by the vibrometer. Frequencies ranging from 0 to 2KHz were used in the experiment to simulate vibration disturbances commonly incurred in the environment.
3. Experimental results
The measured results of the lens operating under several different conditions were shown in Fig. 3 . Considering the working principle of the currently adopted interferometer based laser Doppler vibrometer (LDV), under the effect of vibration, the resultant frequency shift ∆f can be expressed by
In current testing configuration, the laser is first transmitted through the whole lens and then focused onto the membrane. If the membrane is vibrated under, the resultant OPD variation should be, where n is the refractive index of the medium of lens body, namely water (n = 1.33) in current case. As a result, the variation gradient of the OPD can be described by
Considering the fact that the vibration velocity of the membrane is 2πfX, it can be seen that the measured velocity data V with LDV is actually n times that of the real value in current case. As a result, during the experiment, the real vibration velocity spectra vr(f) is first calculated from the measured data v(f) of LDV. Subsequently, the real displacement spectra of vibration X(f) can be obtained by
Figures 3(a) and 3(b) demonstrated the measured frequency response spectra of the vibration amplitude obtained from the same lens working with intracavity pressures of 5603Pa (resultant focal length of 18mm) and 6821Pa (resultant focal length of 17.28mm), respectively.
In both cases, the vibrations were activated using three different sound levels, namely 50dB, 66dB and 71dB, which correspond to 6.3mPa, 39.9mPa and 71mPa acoustic pressure at the membrane surface. It can be clearly seen that there are several distinct resonant peaks appearing within the 0~2kHz frequency range and the amplitudes of all the peaks are monolithically increased with the increasing sound level. The amplitude variation of one peak was selectively shown in the insets and the slopes were found to be 5.48 × 10−4 μm/mPa and 4.21 × 10−4 μm/mPa for the lens working under lower (5603Pa) and higher (6821Pa) intracavity pressure, respectively.
Figure 3(c) shows the vibration spectrum of the lens under varying intracavity pressures, in which all the vibrations were driven by the same sound level of 71dB. From the results, it can be seen that the first several peaks in the lower frequency range (< 500Hz) were found to be nearly constant (no frequency shift) in spite of the pressure variation in the lens cavity. These peaks can be treated as the resonances from the lens base as well as its supporting frame considering that the pressure change in the lens cavity has less/negligible effect on these vibration characteristics. In contrast, in the higher frequency range (>500Hz) as shown in Fig. 3(d), there are three main peaks that demonstrated clear frequency shift with the change of the pressure.
Since the tension/stiffness of the lens membrane is dependent on the pressure induced deformation, which will eventually modulate its mechanical response, it can be seen that these peaks were resonance response of the vibrated membrane in various resonant modes. Figure 3(e) shows the frequency shifts of these three resonant peaks with the change of pressure in the lens cavity. It is clear that all of the three resonant peaks were simultaneously shifted to higher frequency with the pressure increasing from 4773Pa to 6821Pa, in which the frequencies start from 592Hz, 1311Hz and 1702Hz, and finally shift to 658Hz, 1436Hz and 1854Hz for the 1st, 2nd and 3rd resonant peaks, respectively.
To analyze the mechanical response of the liquid filled lens, we can approximate the surface of the liquid-filled lens as an elastic membrane with clamped boundary. When liquid is introduced into the lens cavity, the resultant intracavity hydraulic pressure Pintra will deform the membrane and at the same time raise the tension σ in the membrane, which can be expressed by 
Under the effect of tension, the resonant frequency of the lens membrane can be described by 
Since the condition for higher order vibration is very complicated and beyond the scope of this paper, currently only the case of vibration under fundamental mode is theoretically analyzed for comparison. Substituting the corresponding values of lens structure dimension and material properties into Eqs. (4) and (5) and supposing the damping related coefficient to be 0.6, the frequency shift with the variation of the pressure in the lens cavity can be obtained as shown in Fig. 4 together with the measured results. From the figure, it can be seen that good agreement between theoretical prediction and the measured results can be found.
From the results shown in Fig. 3(c), it is obvious that among the three peaks, the vibration amplitude of the first peak (belonging to the membrane vibrated under the fundamental mode) is found to be the largest, demonstrating the most distinct effect on the lens performance. As shown by previous studies [21, 22], the fundamental mode of similar structure is the out-of-plane deformation of the whole membrane. As a result, in current arrangement, the largest instability of liquid-filled lens induced by external perturbation should be the fluctuation of its focal length. From the results shown in Fig. 3(b), if the lens is exposed to external perturbation with 1kPa pressure, the induced vibration amplitude of the membrane can be calculated to be around 0.4mm (4.21 × 10−4µm/mPa × 1kPa). Considering the commonly used spherical shape approximation of lens , this will cause 1.2mm, namely 6.95%, fluctuation of the focal length. At the same time, the lens profile will also be affected by the vibration, thus introducing image aberration. Due to the limitation of current LDV, this effect will be studied in the future characterization of lens optical performance.
Vibrating under the fundamental mode, when external force (such as the acoustic pressure Pacous in current experiment) is applied to the already stretched membrane, the induced deformation can be calculated as 
Similarly, from both theoretical analysis (Eq. (6)) and experiment result (Fig. 3(a) and 3 (b)), it can be seen that stronger excitation will cause larger vibration irrespective of the cavity pressure. Meanwhile, given the same external excitation, lens working under shorter focal length demonstrates smaller deformation (see insets in Fig. 3(a) and 3(b)) and thus higher stability.
When designing a liquid-filled lens for real applications, the lens diameter is usually determined by the application requirements. Given this condition, in order to achieve higher operation stability, it is undoubtedly much more desired to design lens with its resonant frequency far away from the frequency spectrum of the perturbation during the whole operation range. Considering the cases that the perturbation spans a wide range and sometimes it’s difficult to be predicted, the most straightforward and reasonable strategy is to adopt membrane with larger thickness and higher Young’s modulus so as to make the membrane stiffer. These can be easily achieved by controlling process parameters such as reducing the spinning speed (for larger thickness) and increasing the mixing ratio of the curing agent (for higher Young’s modulus). However, it works at the expense of higher requirement on the actuator (provide larger actuation capability) for achieving focal length tuning. Therefore, a tradeoff should be made to balance both of these requirements with respect to particular cases. Another strategy is the adoption of liquid with higher viscosity than water, such as oil, which will increase the damping effect and dissipate more energy from environment perturbation, thus reducing the membrane vibration and improving the lens stability. From experimental results, it can also be seen that the vibrations from the lens base and the fixture frame will also affect lens stability. Through designing stiffer lens base (prefer smaller thickness and higher Young’s modulus) and more stable fixing structure, this effect can be largely eliminated.
In this paper, the dynamic mechanical stability of the liquid-filled lenses under external perturbation is studied. Through the use of acoustic wave as the broad band disturbing source, the frequency response of the lens working under different conditions within the 0-2kHz frequency range has been successfully characterized. From the experimental results, a clear resonance shift to higher frequency and reduced vibration velocity as well as magnitude can be found with the increasing of the pressure in the lens cavity, namely shortening the focal length, which agrees well with the theoretical predictions. Combining experimental and theoretical analysis, the focal length fluctuation of the liquid-filled lens is considered to be the main concern caused by its dynamic mechanical stability and some technical solutions have been provided accordingly. To the best of our knowledge, it’s the first time that the mechanical stability of the liquid-filled lenses was experimentally reported. In the future, we will focus on quantitatively testing the resultant effect of the mechanical stability on the lens profile as well as the resultant final optical performance such as image aberration. All of these results will undoubtedly provide useful references and help researchers design their own liquid-filled lenses for particular applications.
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