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A compact, high-speed variable-focus liquid lens using acoustic radiation force is proposed. The lens consists of an annular piezoelectric ultrasound transducer and an aluminum cell (height: 3 mm; diameter: 6 mm) filled with degassed water and silicone oil. The profile of the oil-water interface can be rapidly varied by applying acoustic radiation force from the transducer, allowing the liquid lens to be operated as a variable-focus lens. A theoretical model based on a spring-mass-dashpot model is proposed for the vibration of the lens. The sound pressure distribution in the lens was calculated by finite element analysis and it suggests that an acoustic standing wave is generated in the lens. The fastest response time of 6.7 ms was obtained with silicone oil with a kinematic viscosity of 100 cSt.
©2010 Optical Society of America
1. Introduction
Cameras with rapid responses are strongly desired since they enable an object moving rapidly in the axial direction to be recorded without defocusing. Additionally, they can obtain images with a large depth of fields by sweeping the focal point in the axial direction and integrating the images obtained [1]. Most built-in cameras in mobile electronic devices are bulky because they have a camera lens, and an actuator and gearing system, which move the lens in the axial direction. Variable-focus liquid lenses with no mechanical moving parts have been investigated by several researchers [2–6]. They have the potential to be more compact and have faster responses than conventional mechanical lens. Most liquid lenses utilize electrowetting, which is a phenomenon that alters the wetting properties of a metal as a result of ions being generated at the metal surface when an electric potential is applied. This enables the contact angle between the liquid and the metal substrate in a lens to be controlled. This ability to change the contact angle allows the lens surface to be deformed and the focal point of the lens to be varied. These lenses have relatively long response times on the order of tens of milliseconds that depend on the physical properties of the metal substrate and the liquid [2]. López et al. developed an oscillating liquid lens that changes its shape and focal length when excited by sound and that functions as a variable-focus lens [7]. Although this lens is capable of rapid focusing, it continuously oscillates in synchronization with the driving sound pressure so that its focal point is continuously swept in the axial direction. Oku et al. developed a high-speed liquid lens that employs a piezoelectric actuator [8]. This lens is deformed by using the actuator to alter the liquid pressure in the lens. The liquid pressure is amplified through Pascal’s principle. Consequently, the lens requires a magnification mechanism, which makes the camera rather bulky. We have been investigating liquid lenses that use acoustic radiation force [9]. Acoustic radiation force is a static force generated by a difference in the acoustic energy densities of two different media, which depends on the speed of sound and the density of the two media [10–15]. When an ultrasound beam irradiates the interface between two immiscible liquids, the interface is deformed by acoustic radiation force [16,17]. Our lens had a response time of 40 ms, which is comparable to that of mechanical lenses. However, it was 30 mm long, which is rather bulky for portable electronic devices, since it requires a concave ultrasound transducer to focus the ultrasound beam and enhance the acoustic radiation force at the lens surface. The present paper discusses a more compact, high-speed variable-focus liquid lens that uses acoustic radiation force and has no moving mechanical parts.
2. Configuration of the liquid lens
Like conventional liquid lenses, our variable-focus liquid lens utilizes the refractive index difference between two immiscible liquids and its focal point can be translated along the optical axis by deforming the interface between the two liquids. Figure 1 shows the configuration of the liquid lens. The lens consists of a cylindrical aluminum cell (inner diameter: 3 mm; outer diameter: 6 mm; height: 3 mm) filled with two immiscible liquids (water and silicone oil) and a piezoelectric lead zirconate titanate (PZT) ring (outer diameter: 4 mm; inner diameter: 2 mm; height: 1 mm; resonance frequency: 2 MHz) polarized in the height direction. The PZT ring was attached to the bottom of the cell by epoxy. Two circular quartz plates were installed on the center of the PZT ring and the top of the lens to seal the liquids and to allow light to propagate through the lens in the axial direction [Fig. 1(b)]. In the prototype, degassed water (refractive index n 1: 1.333) and silicone oils with various kinematic viscosities (KF-96-2cs to KF-96-1,000cs, Shin-Etsu Silicone, Japan; kinematic viscosity: 2 to 1000 cSt; specific gravity: 0.873 to 0.970; refractive index n 2: 1.391 to 1.403; speed of sound: 931.3 to 987.3 m/s) were used to prevent cavitation bubbles being generated in the lens. An appropriate volume of degassed water was deposited on the reentrant on the bottom glass using a micropipette so that the hemisphere of water that formed was approximately 1.5 mm high. The shape of the water drop differed slightly in each lens due to different interfacial tensions with the different silicone oils. The interface profile changed little due to van der Waals adsorption between the hemisphere of water and the quartz plate when the lens was inverted. Using ultrasound in the megahertz range prevents cavitation bubbles being formed and the water and oil forming an emulsion [18].
3. Theoretical model for vibration of the lens
The response time of the lens depends on both the interfacial tension and the kinematic viscosities of the liquids. When the oil-water interface is statically deformed, the acoustic radiation force acting on the interface and the restoring force (equal to the product of the interfacial tensions of the two liquids and the interface displacement) are balanced. Sakai et al. found that the time constant of the transient response is proportional to the viscosities of the two liquids and inversely proportional to the interfacial tensions [19]. We assume that free vibration of the hemispherical water droplet in the liquid lens can be expressed by a simple spring-mass-dashpot model (see Fig. 2 ). The vibrational displacement of the top position of the droplet x satisfies the following differential equations:
where m is the effective mass, γ is the resistance related to the viscosity of the liquids, k is the spring constant related to the interfacial tension, α ( = γ/2m) is the vibrational damping coefficient, and ω 0 ( = ) is the free angular frequency. When the inertia is dominant for low-viscosity oils and α < ω 0, the solution of Eq. (1) can be expressed aswhere ω ( = ) is the damped angular frequency. Strani and Sabetta [20] and Watanabe [21] theoretically investigated the free vibration of a liquid droplet that is much denser than the surrounding media and is partially supported by a solid rod. From the exact solution for the fundamental vibration mode of the droplet, Shiokawa’s group derived the spring constant k to be 4π2 σ (where σ is the surface tension of the liquid) and found that it agreed well with experimental results [22]. As with the theoretical model of Strani and Sabetta [20], if it is assumed that the bottom part of a 0.5-mm-high water droplet shown in Fig. 1(c) is fixed, the mass m can be expressed aswhere mw is the mass of the hemispherical water droplet, mo is the additional mass in the surrounding oil, R is the water droplet radius, ρw is the density of water, ρo is the density of the oil, and Vo is the volume of the surrounding oil, which acts as an additional mass. When the resistance term is dominant for a high-viscosity oil, a non-oscillatory response will appear and the resistance γ can be expressed as γ = kτ = 4π2 στ (where τ is the time constant). If we use the duration of the transient response as the response time of the lens, the kinematic viscosity that minimizes the response time can be derived from under the critical damping condition. To fabricate a high-speed lens, it is critical to miniaturize the lens since reducing the mass of water will reduce the time constant of the lens under the critical damping condition.4. Finite element analysis of the sound pressure field in the lens
In the configuration of the designed lens, both an annular PZT element and transparent glass in the center should be employed since the optical axis lies along the central axis. The oil-water interface is deformed by the acoustic radiation force from the transducer. The interface shape depends on the sound pressure distribution in the lens. To increase the deformation of the interface along the optical axis, the sound pressure amplitude along the central axis needs to be increased. It is difficult to measure the sound pressure distribution using a conventional ultrasound probe (e.g., a polyvinylidene fluoride needle hydrophone) in this liquid lens since the diameter of a typical hydrophone (approximately 1 mm) is equivalent to or larger than the wavelength of the ultrasound in water in the megahertz range (0.75 mm at 2 MHz). A fiber optic probe with a diameter smaller than the ultrasound wavelength [23] is suitable for performing measurements. Instead, the sound pressure distribution was predicted by finite element analysis (FEA). The sound pressure distribution in such a small region can be precisely calculated by FEA [24,25]. Figure 3 shows the FEA simulation model. A liquid lens with the same configuration as that shown in Fig. 1 was modeled. To reduce the total computational cost, one-quarter of the lens was modeled by applying symmetric boundary conditions in the r−z plane. The average mesh size was 0.1 mm. At the interface between the liquids and the solid, the vibrational displacement of the solid elements was converted into a sound pressure in the liquids. The boundary condition on the outer surface of the lens was perfect reflection of sound waves. A continuous sinusoidal electrical signal was input to the PZT electrodes. The sound pressure distribution in the lens in the steady state was calculated by analyzing the interactions between the piezoelectric, structural, and acoustic elements using FEA. The acoustic radiation force, the interfacial tension, and the liquid viscosities were not taken into consideration and the profile of the oil−water interface was not affected by the acoustic radiation force throughout the calculation. Figure 4 shows the radial distribution of the vibrational displacement amplitude on the bottom of the lens and the sound pressure distributions in the r−z and r−θ planes at a driving frequency of 1.62 MHz. Figure 4(b) shows that a bending vibration converted from vibration of the PZT ring was generated along the central quartz plate and that the vibration amplitude along the central axis is larger than that of the PZT; this assisted in generating a large sound pressure along the central axis. Although a complex acoustic standing-wave field is generated in the axial and radial directions, a large sound pressure amplitude is produced on the central axis of the lens. The quartz plate thickness will also affect the sound pressure distribution.
Fig. 4 FEA results. (a) Schematic layout of distributions (b), (c) and (d). (b) Vibrational displacement amplitude in the radial direction on the bottom of the lens. Sound pressure distributions in the (c) r−z and (d) r−θ planes at z = 2 mm.
5. Focusing performance of the lens
The characteristics of the prototype variable-focus lens were investigated. The profile of the oil−water interface could be changed by controlling the acoustic radiation force from the transducer. It was difficult to observe the oil−water interface using a laser Doppler vibrometer because a sufficiently strong reflected light signal could not be obtained from the interface. In this paper, the change in the interface shape when varying the transducer driving voltage was observed by optical coherence tomography (OCT; IV-2000, Santec, Japan), which is an optical interferometric technique that images variations in the refractive index [26]. The region of interest in the OCT measurements was a 2 × 2 × 3.8 mm3 volume near the center of the lens. Figure 5 shows radial profiles of the oil−water interface excited by five different driving voltages in the range 0 to 51 V at a frequency of 1.62 MHz. The kinematic viscosity of the silicone oil was 200 cSt and a radial distance of 0 mm corresponds to the central axis. When the driving voltage of the transducer was increased, the oil−water interface on the central axis moved toward the water side due to the acoustic radiation force. The direction of the interface movement is from the oil side with a higher acoustic energy density (lower density and a lower speed of sound) to the water side with a lower acoustic energy density (higher density and a higher speed of sound). This axisymmetrically deformed oil−water interface acts as a variable-focus lens. Figure 6 shows the relationship between the input voltage to the transducer and the displacement of the interface along the central axis relative to its position when no ultrasound radiation is applied. The maximum displacement of the lens was 0.27 mm at an input voltage of 51 V, and the gradient of the curve was increased with increasing input voltage since the acoustic radiation force is proportional to the square of the sound pressure amplitude. However, when the input voltage exceeded approximately 51 V, cavitation bubbles and acoustic streaming were generated and the oil-water interface did not stabilize with time.
Fig. 5 Radial profiles of the oil-water interface observed by OCT at five different driving voltages.
Fig. 6 Relationship between the input voltage to the transducer and the interface displacement at the center relative to its position when no ultrasound radiation is applied.
The beam profile of the light transmitted through the lens was investigated. The incident light was refracted at the oil−water interface and the beam profile of the transmitted light can be altered by varying the oil−water interface. The transmitted light distribution was calculated by ray tracing using commercial ray-tracing software (TracePro, Lambda Research Corp., MA). Figure 7 shows the simulation models and the computed results for cases when the lens was excited by supplying input voltages in the range 0 to 51 V. A 2-D model was used in the simulation and the OCT experimental results (see Fig. 5) were used for the shape of the oil−water interface. Based on the lens profiles shown in Fig. 5, the lens aperture is approximately 0.2 mm or smaller. Incident light with a beam width of 0.2 mm passed through the lens to the right-hand side. The 1-mm-thick quartz plate at the lens surface was accounted for in the simulation. The refractive indices of water, oil, and glass were taken to be 1.333, 1.400, and 1.476, respectively. The profiles of the transmitted light varied with deformation of the oil−water interface. The beam was focused when input voltages of 44 and 51 V were applied. Here, we determined the beam width in the manner shown in Fig. 7. Figure 8 shows the beam width as a function of axial distance for several input voltages. An axial distance of 0 mm corresponds to the lens surface. For input voltages of 44 and 51 V, the beam widths have minimum values at distances of 4.1 and 1.7 mm from the lens surface, respectively; these positions correspond to the focal point of the lens. For input voltages of 0, 26, and 35 V, the transmitted light diverged due to the convex surface of the lens and there was no focal point. These results reveal that increasing the input voltage reduces the focal length. The focal point moved to infinity when the input voltage was reduced from 44 to 35 V. The relatively large beam widths near the focal point for input voltages of 44 and 51 V indicates that the liquid lens suffers from spherical aberration. Figures 9(a) and 9(b) show the spherical aberrations of the lens, which is the relationship between the focal point and the radial position of a ray for input voltages of 44 and 51 V, respectively. A radial position of 0 mm in the vertical axis corresponds to the center of the lens on the optical axis. If a lens does not have spherical aberration, a straight line paralleled to the vertical axis will appear at the focal point. Figure 9(b) (for 51 V) shows a typical curve for spherical aberration: the focal length decreases as the distance between the central axis and the radial position of the ray increases. When the input voltage was 44 V, the shape of the curve differed from that for an input voltage of 51 V, especially the center of the lens, and most of the rays in the radial direction were focused at an axial distance of 2.3 mm from the lens surface (although the focal length estimated from the beam width was 4.1 mm, as shown in Fig. 8).
Fig. 9 Relationship between the focal point and the radial position of a ray for input voltages of (a) 44 and (b) 51 V.
The liquid lens was employed experimentally as a variable-focus lens. The laser beam passed through the lens perpendicularly, as shown in Fig. 1(b), and the incident light was focused by exciting the lens. A He-Ne laser beam with a width of 0.2 mm was used as the incident light. The radial distribution of the light intensity was measured by scanning a photodetector with a pinhole aperture of 30 µm at a distance of 3 mm from the lens surface. Figure 10 shows the radial intensity distribution of the transmitted laser beam when the lens was excited with voltages of 0 and 35 V. The laser beam profile changed on applying an input voltage; the full widths at half maximum were calculated to be 0.255 and 0.215 mm for input voltages of 0 and 35 V, respectively.
Fig. 10 Radial intensity distributions of the transmitted laser beam produced by a liquid lens for input voltages of 0 and 35 V.
6. Dynamic response of the lens
The transient response of the liquid lens was investigated. The motion of the oil−water interface was continuously observed using the M-mode of OCT. Figures 11(a) and 11(b) show the transient responses of the oil−water interface displacement along the central axis measured by OCT when the ultrasound radiation was turned on and off, respectively. The interface was observed at the central axis. The kinematic viscosities of the oil were 10, 100, and 1000 cSt. For all three oils, the input voltage was controlled to give a displacement of 0.2 mm since the acoustic radiation force depends on the physical properties of the oil. In Fig. 11(a), the input voltage was applied at t = 0 and the interface began to move almost immediately because the acoustic pressure field at 1.62 MHz has a submillisecond transient time, which is considerably smaller than that of the interface. The interface reached to a steady state after a transient state that lasted several tens of milliseconds. These experimental results reveal that a shorter time constant for the transient response can be obtained by using a less viscous oil, although damped oscillation was observed with an oil with a viscosity of 10 cSt. These results reveal that there is an optimal kinematic viscosity of the oil that minimizes the response time, as stated in §3. When the ultrasound was switched off [see Fig. 11(b)], the same tendencies were observed in the time responses as when it was switched on. However, the waveforms are different and simple damped oscillation for the 10-cSt oil was observed since the restoring force acted without the acoustic radiation force.
Fig. 11 Transient responses of the oil−water interface for kinematic viscosities of 10, 100, and 1000 cSt when ultrasound radiation was switched (a) on and (b) off.
If the silicone oil with a kinematic viscosity was 2 cSt, the damped oscillation in the response curve (see Fig. 11) could be clearly observed when the ultrasound was switched off; the damped angular frequency ω and the vibrational damping coefficient α were respectively estimated to be 384 rad/s and 41.4 from the waveform. From these values the free angular frequency ω 0 is calculated to be 386 rad/s and the effective mass m ( = ) is estimated to be 9.61 × 10−6 kg (here, the interfacial tension between water and the silicone oil σ of 36.3 mN/m was used). From Eq. (3), mw and mo are respectively calculated to be 2.09 × 10−6 and 7.52 × 10−6 kg and the surrounding oil with a volume Vo of 8.61 × 10−9 m3 (which is 3.6 times larger than that of the hemispherical water droplet) is predicted to act as an additional mass. This result implies that the volume of the surrounding oil can be decreased and the liquid lens can be downsized with the same volume of the water droplet.
Figure 12 shows the relationship between the kinematic viscosity of the silicone oil and the time constant of the transient response τ. With or without damped oscillation of the oil−water interface, the time constant τ corresponds approximately to the time until the interface reaches 63.2% of the position of the steady state when ultrasound radiation is switched off. With increasing viscosity, the time constant τ asymptotically approaches a proportional relationship with the viscosity η (τ ∝ η). It should be noted that the silicone oils used in the experiment have slightly different interfacial tensions and densities. With decreasing viscosity, damped oscillations appeared so that the effect of the inertial term cannot be neglected [27] and the result clearly deviate from the line τ ∝ η. Figure 13 shows the relationship between the kinematic viscosity of the oil η and the resistance γ calculated from Fig. 12 for the non-oscillatory response. In this region, the resistance γ is proportional to η 0.55. Considering the response time of the lens, the optimal kinematic viscosity is estimated to be 54 cSt from the experimental results shown in Fig. 13 and the theoretical critical damping condition of. Figure 14 shows the relationship between the kinematic viscosity and the response time of the lens. The optimal kinematic viscosity that minimizes the response time was 100 cSt, which is relatively close to the predicted value of 54 cSt, and the shortest response times of 8.3 and 6.7 ms were achieved when the ultrasound was turned on and off, respectively. These results imply that the focal point of the lens can be varied from infinity to 4.1 mm within 6.7 ms.
Fig. 12 Relationship between the kinematic viscosity of the silicone oil and the time constant when ultrasound radiation is switched off.
Fig. 13 Relationship between the kinematic viscosity of the silicone oil and the resistance in the free vibration model.
Fig. 14 Relationship between the kinematic viscosity of the silicone oil and the response time of the lens when ultrasound radiation is switched on and off.
7. Conclusions
A compact, high-speed liquid lens using acoustic radiation force was developed. A 4-mm-high and 6-mm-diameter liquid lens was produced that consisted of a hemispherical water droplet surrounded by silicone oil, a PZT ring, and an aluminum cell. The oil−water interface can be deformed by acoustic radiation force and it acts as the surface of a variable-focus lens. The dynamic response of the lens was modeled using a spring−mass−dashpot model. This model predicts that there is an optimal kinematic viscosity of the silicone oil that minimizes the response time of the lens. The change in the profile was measured by OCT. Profiles of the laser beam transmitted through the lens were calculated by ray tracing. The beam profile of the transmitted laser beam could be changed and focused by applying an input voltage so that the liquid lens functioned as a variable-focus lens. The shortest response times were obtained for the silicone oil with a kinematic viscosity of 100 cSt; they were 8.3 and 6.7 ms when the ultrasonic radiation was switched on and off, respectively.
Acknowledgment
This work was partially supported by a research grant from Japan Science and Technology Agency, and we acknowledge the provision of ray-tracing software from Lambda Research Corp. and FET, Inc.
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