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This paper proposes a variable-focus lens with 1-kHz bandwidth. The lens transforms its shape rapidly using the liquid pressure generated by a piezo stack actuator. This mechanism also includes a built-in motion amplifier with high bandwidth to compensate for the short working range of the piezo stack actuator. Prototypes have been developed to validate the proposed design. A 1-kHz bandwidth of the lenses was confirmed by measuring the frequency responses. Refractive power ranging from -1/167 to 1/129 mm-1 and a maximum resolution of 12.3 cycles/mm were attained.
©2004 Optical Society of America
1. Introduction
High-speed focusing is required in many application fields as a result of the recent development and widespread use of optical devices. For example, rapid axial scanning of the focal plane is important for confocal scanning microscopes to acquire three-dimensional information of objects at high speed. High-speed focusing can also realize three-dimensional displays using axial scanning of the image plane position. In these application fields, since millisecond-order scanning is required, we assume that 1-kHz bandwidth is adequate for our proposed high-speed focusing device.
However, no previous focusing mechanisms have yet attained a 1-kHz bandwidth. For example, the axial tracking mechanism of optical disks, which is one known fast-focusing system, has a first resonant frequency of around 100 Hz [1].
A lens that realizes variable focusing by changing its shape is considered a key device to achieve high bandwidth. Although several studies have reported such variable-focus lens structures [2, 3] and micro variable-focus lens arrays [4, 5], few have aimed at achieving high bandwidth. Day et al. have developed a variable-focus lens based on the modification of the wetta-bility of a surface [6]; this lens took 0.1 s to scan the entire focal-length range from minimum to maximum. Kaneko et al. have developed variable-focus lenses with deformable glass surfaces [7] using a PZT bimorph actuator; however, their prototype was limited to a bandwidth of 150 Hz.
In this paper, we propose a new structure of a variable-focus lens that can realize 1-kHz bandwidth.
2. Focusing mechanism
The proposed lens focuses rapidly by changing the lens shape. Figure 1 shows its structure and focusing mechanism. The lens surfaces are composed of two parallel transparent disks, at least one of which is deformable. The space between the disks is filled with a transparent liquid. One disk is deformed by the pressure difference between the liquid and the atmosphere. Convex and concave lenses can be realized by increasing and decreasing the pressure to push out or pull in the disk, respectively. The pressure of the liquid is generated by an actuator which thrusts against an elastic plate called a cylinder. By deforming the cylinder, pressure is applied to the liquid and transmitted to the lens surfaces.
The elastic lens surface and the cylinder must be designed to have a first resonance frequency of transverse vibrations that is sufficiently higher than 1 kHz. This is because a lower resonance frequency causes resonance vibrations, which makes it difficult to rapidly control the focal length.
A bandwidth of more than 1 kHz is also required for the actuator. One suitable actuator is a piezo stack actuator, which has a high bandwidth (up to ~ 10 kHz). However, the working range of the piezo stack actuator is too short (~ 10 μm) for actuating the lens surface directly. To solve this problem, the lens structure includes a built-in motion amplifier making use of the incompressibility of the liquid: The cross-sectional area of the cylinder S is designed to be several tens of times larger than that of the lens surface s, and therefore the distortion of the lens surface is approximately S/s times that of the cylinder.
3. Theoretical analysis of the dynamic and static deformation of the elastic surfaces
Analysis of the static and dynamic behavior of deformable parts, such as the lens surfaces and the cylinder, is essential to properly design the lens. Thus, a theoretical analysis of the deformation will be explained here to prepare for further discussion on the design optimization in the next section.
Elastic circular plates whose edges were rigidly fixed to the container of liquid were adopted as the lens surface and the cylinder. Hence, the lens surface is modeled as a disk of radius a and thickness d, formed of a material of Young’s modulus E, density ρ, and Poisson’s ratio ν; the periphery of the disk is completely fixed. A schematic diagram of this model is shown in Fig. 2.
According to the theory of vibration [8], the natural frequency ω of such a disk is
Here, m and n indicate the vibration mode. The parameter k m,n is the solution of
where I m is the m-th modified Bessel function of the first kind, and J m is the m-th Bessel function of the first kind. A vibration mode with m = 0,n = 1 corresponds to the first natural vibration. Note that k m,n is proportional to 1/a because it is a solution of Eq. (2).
When a static pressure P is applied to one surface of the disk, the deformation z = R(r) is denoted by the following polynomial:
where D denotes the flexural rigidity, given by:
4. Optimizing the design of the proposed lens
A high natural frequency of transverse vibrations is preferable for the elastic plates forming the lens surface and the cylinder. This is because, resonance vibrations easily occur in a disk with a low natural frequency. Consequently, complicated feedback control is needed to control the lens at high-speed, which increases the cost and complexity of the system. At the same time, the lens is also required to have a focal length of sufficient range. Hence, the lens surfaces and the cylinder need to have not only a high first natural frequency but also high elasticity. However, a high elasticity is not consistent with a high natural frequency. Therefore, the trade-off between these properties should be considered.
To evaluate the elasticity of the disk, we define a new parameter S, which is the ratio of the curvature at the center of the disk to the static pressure of the liquid. The refractive power of the lens, which is the inverse of the focal length, is related to the curvature of the disk. Assuming a plano-convex or plano-concave lens, the relation is denoted by 1/f = (n r -1)c, where n r is the refractive index of the material that forms the lens. Thus, S represents the refractive power per unit pressure; moreover, large S indicates large elasticity.
According to Eq. (3), the curvature at the center of the disk c is
We can obtain S using the relations for c and D in Eq. (5) and Eq. (4), as follows:
Table 1. Relations between parameters
Table 1 summarizes how the first natural frequency ω 0,1 and S depend on each parameter of the disk. The disk can be properly designed by considering these relations.
As for the disk dimensions, that is, radius a and thickness d, a thin, small disk is better than a thick, large disk. To explain the reason, consider how to attain a natural frequency four times higher for a given disk of radius a 0, thickness d 0, first natural frequency ω 0 and elasticity S 0. Assuming a change in radius to new radius a, a is required to be a 0/2 to attain natural frequency four times higher. Inserting a = a 0/2 in each factor in table 1 yields ω = 22 ω 0 and S r = S 0/22 = S 0/4. In contrast, if we modify the thickness to new thickness d, d is required to be 4d 0. Inserting this in the relations in table 1 yields ω = 4ω 0 and S t = S 0/43 = S 0/64. According to the result, S r is 16 times larger than S t. This indicates that modifying the radius can attain a much higher elasticity than modifying the thickness. Hence, to attain a desired first natural frequency, modifying the radius is better than modifying the thickness. Therefore, a lens with smaller lens radius can attain a given refractive power with less liquid pressure.
Fig. 3. Materials [4][9][10]: Glasses were arbitrarily selected from a thin sheet glass catalogue (Corning Inc.). Metals are plotted because they can be used as a material for the cylinder.
From these results, we can also derive adequate materials for the disk. Figure 3 shows a scatter plot of various materials depending on their first natural frequency and elasticity. Materials plotted at the upper right, such as glasses and plastics commonly used for lenses, are adequate for the disk. This is one advantage of the proposed lens, because common lens fabrication techniques and materials, such as optical coatings can be used.
5. Experimental results
5.1. Fabrication of the prototypes
Based on the above discussion, two kinds of prototypes were fabricated: prototypes #1 and #2. Prototype #1 was fabricated simply to determine whether the proposed structure could achieve a 1-kHz response. Thus, its structure was designed to be simple and easy to fabricate, but not durable. After confirming the response speed of prototype #1, prototype #2 was designed to measure more detailed characteristics, such as surface profile and resolution. Mechanical specifications of these prototypes are shown in Table 2.
Table 2. Mechanical specifications
Prototype #1 had a body composed of a silica glass tube as a rigid container, and it had two 5-mm diameter lens surfaces placed facing each other at the opposite sides of the tube. A schematic diagram of the structure is shown in Fig. 4(b). The inner diameter of the glass tube was 27 mm and the lens thickness was about 34 mm. One lens surface was a thin elastic silica glass disk with a natural frequency of 7.2 kHz, which was sufficiently higher than the required 1 kHz. The other, a rigid surface, was a 1.0-mm thick silica glass plate. The elastic lens surface consisted of a 20-μm thick and 5.5-mm diameter glass plate bonded on a 1.0-mm thick glass plate having a 5-mm diameter hole at the center, as shown in the inset of Fig. 4(b).
5.1.1. Prototype #1
Fig. 4. Prototype #1 :(a) photograph, and (b) schematic figure of the structure and fabrication process
One end of the glass tube was covered with a 5-mm thick acrylic plate with two inlets for the transparent liquid, and the other end was covered with a 0.5-mm thick stainless steel (SUS304) deformable plate with a diameter of 27 mm, which functioned as the cylinder. The liquid inlets were connected to conventional stopcocks of the type normally used in chemical experiments.
The container was filled with a fused silica index-matching oil (Cargille laboratories) whose refractive index matched that of fused silica (1.4587 at a wavelength of 5893Å).
Prototype #2 was designed to overcome a drawback encountered in prototype #1: Through frequency response measurements of prototype #1, it was found that the structure of the elastic lens surface was very fragile. In fact, prototype #1 was broken just after the frequency response measurement; it lacked sufficient durability to measure various characteristics, such as the lens surface deformation shape, resolution, etc.
5.1.2. Prototype #2
Fig. 5. Prototype #2:(a) photograph, and (b) schematic diagram of the structure and fabrication process.
To solve this problem, the elastic lens surface of prototype #2 was designed to be much more durable than that of prototype #1; its structure was as follows. The elastic lens surface consisted of one 40-μm-thick silica glass plate sandwiched between two identical 480-μm-thick silica glass substrates. The two substrates had a circular hole 7-mm in diameter at identical positions so that the 40-μm-thick glass plate could function as an elastic lens surface with 7-mm diameter and 7.2-kHz natural frequency. A schematic diagram of the fabrication process is shown in the inset of Fig. 5(b). This sandwich structure provided sufficient durability for the elastic lens surface. A silica glass plate of thickness 1 mm was adopted as the other lens surface.
A silica glass disk of diameter 42 mm and thickness 1.41 mm was adopted as the cylinder part; its natural frequency was also 7.2 kHz. As discussed previously, glasses are one suitable material for use as the elastic plates. However, because the center of the cylinder is thrust by a piezo stack actuator, this may cause the glass plate to crack. To prevent this, a thin metal sheet was placed on the center of the glass plate.
The body of prototype #2 was made from stainless steel (SUS304). The lens thickness was designed to be 3.5 mm when the elastic surface was flat. The body had two inlets for the transparent liquid, which were connected to diaphragm valves (Fujikin). Fused-silica matching oil was adopted as the transparent liquid, as in prototype #1.
5.2. Dynamic response
The frequency responses of the prototypes were measured. Both prototypes were actuated by a piezo stack actuator (P-841.10, Physik Instrumente) with a 15-μm working range and an 18-kHz natural frequency. This actuator includes a strain gauge position sensor with high resolution (¡ 1 nm). The input was the displacement of the piezo stack actuator obtained by the position sensor, and the output was the displacement of the center of the deformed lens surface, as measured by a laser displacement meter (LC-2430, Keyence). A personal computer controlled these two instruments. The Bode-plot of prototype #1 shown in Fig. 6(a) demonstrates that the proposed lens maintained a gain of approximately 1 and a phase shift of approximately 0-rad up to a frequency of 1 kHz. The response of prototype #2 shown in Fig. 6(b) is almost the same as that of prototype #1 except for one small reproducible resonance at 340 Hz. We suspect that this is caused by a part supporting the lens and the piezo stack actuator. Figure 7 shows trajectories of the actuator and the lens surface displacements. According to these results, the proposed focusing mechanism has a bandwidth of at least 1 kHz.
Fig. 6. Bode plot of frequency response for prototype #1 (a) and prototype #2 (b). The input was the displacement of the piezo stack actuator, and the output was the displacement of the center of the lens surface. The gain is the ratio of the maximum amplitude of the lens to the maximum amplitude of the piezo stack actuator. In this plot, the gain was normalized by dividing the entire set using a certain value. The phase plot shows the shift between the output and the input.
Fig. 7. Trajectories of the lens surface and the piezo stack actuator displacement during the measurement of frequency response of 1 kHz for prototype #1 (a) and prototype #2 (b).
The viscous resistance of the liquid causes pressure difference between the lens surface and the cylinder. If this difference is larger than the maximum pressure that the actuator can generate, then the actuator can not control the lens surface displacement. We estimated this pressure difference using finite element method (FEM). The maximum pressure difference was estimated at 58 Pa, assuming that 1-kHz sinusoidal input was applied to prototype #2. This value is sufficiently smaller than the maximum pressure of the piezo stack actuator, 7.2 × 105 Pa. Therefore, the viscous resistance is not a problem for prototype #2.
The bandwidth is limited by the first resonance frequency of the disks, such as lens surface and cylinder. For example, the lens surface of prototype #2 had a natural frequency of 7.2 kHz. To prevent resonant vibration, the lens must be operated at lower frequency than 7.2 kHz.
Static lens surface profiles were measured using a laser confocal displacement meter (LT-8110, Keyence) that can measure the displacement of one point. To acquire a two-dimensional surface profile, this sensor was fixed on a computer-controlled xyz-stage and the profile was measured by two-dimensional scanning. Eleven surface profiles were measured by varying the displacement of the piezo stack actuator from 0 to 10 μm. The lens was set to be plano-concave profile at the initial displacement (0 μm) of the actuator. The acquired profiles were subjected to curve fitting using the fourth-order polynomial theoretically described in Eq. (3). Results are shown in Table 3.
5.3. Deformation profile of the lens surface
Table 3. Deflected lens surfaces were fitted with theoretical fourth-order polynomials. The displacement of the piezo stack actuator (δ), coefficients of the fitted curves (r 4, r 2), goodness of the fit (Q), estimated focal length (f), and estimated Seidel spherical aberration (SA) are shown.
To determine the validity of the fourth-order polynomial model, the goodness of fit[11] was calculated by fitting the curve using a certain data set to obtain estimated curve parameters, and then estimating the probability that the actual curve for the measured data set matched the estimated curve. The goodness of fit is expressed by probability Q. The closer the probability Q is to one, the more valid the model is. Table 3 shows the probability Q and focal length f. The value of Q is close to one when the focal length is less than about -200 mm or longer than about 200 mm. In contrast, when the focal length is between these values, the Q value is significantly smaller than one, which suggests that the adopted model is no longer valid.
The focal length and Seidel spherical aberration coefficient were also calculated based on the fitted curves. The Seidel spherical aberration coefficient was estimated using optical CAD software (ZEMAX).
Assuming that the deformation of the cylinder is denoted by Eq. (3), the relation between the refractive power and the displacement of the actuator is
where, f is the focal length, n r is refractive index, ΔC is the actuator displacement, and K is a constant depending on the initial profile when ΔC equals 0. Besides, aC and aL indicate the radius of the cylinder and the lens surface, respectively. According to Eq. (7), refractive power 1/f is a linear function of Δc with slope 4(n r - 1)/.
Refractive power (= 1/f) is plotted against the displacement of the piezo actuator in Fig. 8. This figure shows that the linearity of the refractive power is not good. This bad linearity caused by fabrication errors in the dimensions of the lens surface periphery. Theoretical slope of prototype #2 is 5.4 × 106 m-2, and measured slope was ~1.4× 106 m-2 that was almost a quarter of the theoretical value. We suspect that this is caused by the distortion of a part supporting the lens and the piezo stack actuator.
The measured and fitted surface deflection profiles are shown in Fig. 9. As the focal length become short, the error between the measured and fitted surface profiles shows a saddle-like shape; this indicates that the deflected surface profile is not point symmetric. This asymmetry could be caused by fabrication errors in the dimensions of the lens surface periphery.
Prototype #2 was able to maintain a given surface profile while each surface profile measurement that took about 5 minutes. Furthermore, no fluid leakage was observed. However, we found that the focal length of prototypes depended on its temperature, since the liquid volume expands with increasing temperature. One way to solve this problem is to regulate the temperature to be constant.
5.4. Resolution
The resolution was also measured using prototype #2. A USAF 1951 test chart illuminated by a white light source (halogen lamp) was used to measure the resolution. A CMOS imager was placed at the image plane to acquire the images formed by prototype #2 with various positive focal lengths. The resolution was visually read from the chart image. The results are shown in Table 4. Based on these results, a maximum resolution of 12.3 cycles/mm was achieved. Figure 10 shows the captured image of the standard chart by the CMOS imager when the focal length was 174 mm.
Fig. 9. Measured surface profiles, fitted curves, and error profiles. Surface profiles (a) of prototype #2 were measured at various focal lengths. Each surface profile was fitted with a fourth-order polynomial. Fitted curves are shown in column (b). Column (c) shows the error between the measured surface profile and the fitted curve.
Aberration correction could be achieved without loss of bandwidth using two or more of such lenses. An initial numerical simulation has suggested that the aberrations could be corrected adequately.
Prototype #2 had small N.A., or large f-number, because the ratio of the lens diameter to the focal length was small. The maximum radius of the proposed lens was estimated at 5 mm considering the maximum force of the actuator. Therefore, the lowest f-number of the proposed lens is about 13 at the shortest focal length. This f-number is not sufficient for practical applications. In order to attain small f-number, it is necessary to develop an adequate optical system including the proposed lens.
Table 4. Resolution was measured using a USAF 1951 standard test target at various focal lengths f. Vertical and horizontal resolutions are shown.
Fig. 10. Image of the positive standard test chart using the proposed lens prototype when the focal length was 174 mm.
6. Conclusion
In this paper, a new variable-focus lens with 1-kHz bandwidth is proposed. Based on the discussion of the mechanical characteristics, two kinds of prototypes were designed and fabricated. A bandwidth of 1 kHz of the proposed lens was confirmed by measuring the frequency response. Also, a wide range of refractive power, from -1/167 to 1/129 mm-1, and a resolution of 12.3 cycles/mm were attained.
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