1 Introduction

Optical systems for focal position adjustment are employed in several applications, such as cameras [1, 2], microscopes [3, 4], and laser scanning systems [5, 6]. Among the reported systems, varifocal liquid lenses, which can change their focal length, are widely studied to realize small handheld optical systems, such as mobile phones and endoscopes [7, 8]. One approach to design such a lens is to integrate a micro-sized actuator into a liquid droplet encapsulated by polymer thin films [9, 10]. The focal lengths of these liquid lenses are changed by deforming the shape of the encapsulated droplet using the actuators. Another approach utilizes two different liquids encapsulated in a chamber, and the shape of the liquid interface is deformed by electro-wetting phenomena [11–14] or by the stimuli-responsive expansion of hydrogels [15, 16]. However, these liquid lenses currently have no self-embedded sensing mechanism to detect their own focal positions; therefore, the realization of an optical system for accurate focal position adjustment using only a single varifocal liquid lens for applications such as confocal optical systems is still a challenge.

Here, we report a detailed deformation model of our varifocal liquid lens, and based on this model, we propose a focal length measurement method of this lens in which the capacitance change of the varifocal liquid lens is measured without using extra sensors or other sensing mechanisms. We used the varifocal liquid lens fabricated by Parylene-on-liquid-deposition (POLD), which we previously reported [10]. This liquid lens is composed of a liquid that is encapsulated in a thin polymer film and sandwiched between two electrodes. The shape of the lens can be changed by applying voltages between these electrodes for actuation. This deformation of the lens causes a simultaneous change in its focal length and in the capacitance between the two electrodes. Therefore, the focal length of the lens can be determined by measuring the capacitance change of the two electrodes. We found that the every fabricated lens has a steplike structure with a height of few micrometers at the edge of the lens. Because of this steplike structure, even when the lens is deformed by an applied voltage, a thin liquid film still remained between the two electrodes, hindering the gap between the two electrodes to be reduced further than the structure’s height. According to this detailed observation of the deformation, we found that the change in the lens capacitance was in fact only one-tenth of the value calculated by our simple deformation model previously reported in [10]. By considering the influence of the steplike structure, we designed a capacitance measuring circuit and determined the relationship between the capacitance change and focal length of the varifocal liquid lens. In our proposed method, the capacitance change between the two electrodes is measured by applying a high-frequency sinusoidal signal to the lens. From this characteristic, the focal position of the liquid lens can be changed and measured simultaneously using only the lens itself, without any other sensors. Thus, our varifocal liquid lens can be utilized in optical systems, such as confocal distance sensors, that require highly accurate focal position control. Additionally, to demonstrate the efficiency of the proposed measuring method, we created a confocal distance sensor for laser scanning using our varifocal liquid lens and evaluated its distance measurement accuracy.

2 Focal length measurement of the varifocal liquid lens

2.1 Principle of lens deformation and focal position measurement

Figure 1a shows the design of the varifocal liquid lens and the concept of its deformation. The basic idea of the deformation principle and the details of fabrication are reported in our previous paper [10]. The liquid lens was fabricated by dropping a liquid onto an electrode on a glass wafer that was then encapsulated with a thin polymer layer. After encapsulation, a metal layer, which was thin enough for light to pass through, was evaporated onto this polymer film. Before encapsulation, the liquid droplet assumed the shape of a part of a sphere due to liquid surface tension. After encapsulation, the center part of the liquid droplet still assumed this spherical shape. However, around the edge of the lens, the deposited polymer layer formed a steplike structure of a height less than 10-μm (Fig. 1a).

Fig. 1
figure 1

The concept sketch of the varifocal liquid lens. a An image of the entire structure. The inset shows the deformation at the edge of the liquid lens. b The circuit used to measure the capacitance change of the varifocal liquid lens. Vact = High volt, low frequency: actuate liquid lens, Vmeas = Low volt, high frequency: measure capacitance

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To deform the liquid lens, a voltage was applied between the two electrodes to induce an electrostatic force between them. The magnitude of the electrostatic force is inversely proportional to the distance between the electrodes. Therefore, the electrostatic force is the largest at the edge of the lens, where the distance between the two electrodes is the smallest. This electrostatic force pulls the electrode on the polymer film toward the electrode on the glass wafer, pushing the liquid toward the center of the lens, thus increasing the height of the lens at the center. Because of this deformation, the curvature of the liquid lens increases and its focal length decreases. However, the steplike structure formed around the lens’s edge impedes deformation; a thin liquid layer remained between the polymer layer and the bottom electrode even when the lens was deformed. The height of the liquid layer is similar to the height of the steplike structure.

When the liquid lens is dynamically actuated, hysteresis will exist in its deformation because of the lag in the deformation of the liquid and the polymer film. Therefore, a direct measurement method to detect the focal length of the liquid lens is required. To overcome the influence of the deformation hysteresis, we propose measuring the liquid lens’s focal length based on its capacitance change. When the lens deforms, the distance between the electrodes around the edge of the lens decreases and the capacitance between the two electrodes increases. Therefore, the shape of the liquid lens and its focal length can be determined from the capacitance change of the liquid lens.

To simultaneously measure the capacitance change of the liquid lens and to actuate it, we attached a reference resistance R ref to the liquid lens in series, as shown in Fig. 1b. A synthesis signal composed of an actuation signal V act used to actuate the liquid lens and a measurement signal V meas used to measure the capacitance is applied to this circuit. V act is a high-voltage and low-frequency (≈150 Vp–p and lower than 10 Hz) sinusoidal wave signal, and V meas is a low-voltage and high-frequency (≈3 Vp–p and higher than 100 Hz) sinusoidal wave signal. The magnitude of V meas is set low enough to ensure that it does not deform the liquid lens. In this case, the impedance of the liquid lens capacitance C lens to the low-frequency signal of V act is quite high compared with the reference resistor R ref, such that V act does not influence the voltage across the reference resistor V ref. However, the resistances of C lens and R ref are of the same order of resistance as V meas; thus, C lens can be measured according to V ref.

When the signal is applied to the circuit, the voltage across the reference resistor V ref is described as

$$V_{\text{ref}} = \frac{{R_{\text{ref}} \cdot \omega_{\text{meas}} \cdot C_{\text{lens}} }}{{\sqrt {(R_{\text{ref}} \cdot \omega_{\text{meas}} \cdot C_{\text{lens}} )^{2} + 1} }} \cdot V_{\text{meas}} .$$
(1)

According to Eq. 1, the capacitance of the liquid lens can be described as

$$C_{\text{lens}} = \frac{1}{{R_{\text{ref}} \cdot \omega_{\text{meas}} }} \cdot \sqrt {\frac{{V_{\text{ref}}^{2} }}{{V_{\text{meas}}^{2} - V_{\text{ref}}^{2} }}} ,$$
(2)

where ω meas is the frequency of the applied measurement signal V meas. According to Equation 2, the capacitance of the liquid lens can be calculated from the reference voltage V ref.

The resistance of the reference resistor R ref was 100 kΩ, and the measurement signal V meas was 3.0 Vp–p (500-Hz AC voltage). The voltage across the reference resistance V ref was measured using a lock-in amplifier (LI-570A, NF Corporation). The time constant used to measure V ref was 100 ms, and a low-pass filter with an attenuation slope of −6 dB/oct was used in the lock-in amplifier.

2.2 Fabricated varifocal liquid lens and its optical characteristics

A 170-nm-thick indium-tin-oxide (ITO) layer formed on a 700-μm-thick glass wafer was patterned to form the bottom electrode. Then, a 300-nm-thick CYTOP (CTL-809 M, Asahi Glass Co.) layer was formed and patterned into a ring shape on the ITO electrode. A total of 6 μl of silicone oil (HIVAC F-4, Shin-Etsu Chemical Co.) was dropped inside this ring. The liquid was kept inside the ring-shaped CYTOP layer, and the shape of the liquid became spherical because of the liquid’s surface tension. A 1-μm-thick Parylene-C (Parylene Japan Co.) layer was formed on the liquid surface using the chemical vapor deposition (CVD) method to serve as the polymer film, and a 10-nm-thick Au layer was evaporated to serve as the metal electrode. Because the vapor pressure of silicone oil is low, it did not vaporize even under vacuum pressure. With this Parylene-on-liquid-deposition (POLD) process, we were able to encapsulate the liquid with Parylene-C layer and fabricate the varifocal liquid lens [10]. POLD process, however, slightly modified the shape of the liquid droplet around its edge, shaping the Parylene layer into a steplike structure around the perimeter of the encapsulated lens. The comparison of the liquid profile before and after POLD process shown in Fig. 2 clearly confirms that the liquid shape was modified in such a manner during POLD process.

Fig. 2
figure 2

Deformation of the profile of the liquid edge caused by the POLD

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The fabricated liquid lens is shown in Fig. 3. The fabricated lens had a 10-mm diameter, and its center thickness was approximately 120 μm. The initial focal length of the lens was 153.6 mm, and its initial capacitance was 135 pF. The use of chemically stable materials, such as silicone oil and Parylene-C, permits the liquid lens to maintain its deformation characteristics for a long time.

Fig. 3
figure 3

A photograph of the fabricated varifocal liquid lens

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2.3 Relationship between the capacitance and focal length of the liquid lens

We actuated the liquid lens by applying a static voltage and measured the relationships between its capacitance change and its focal length change.

An image of the liquid lens when it was deformed by an applied voltage of 150 V is shown in Fig. 4a, and its profile at the A–A’ cross-section is shown in Fig. 4b as defined in Fig. 4a. The top polymer layer of the liquid lens came close to its base in an approximately 1-mm-wide ring area around the edge of the liquid lens. However, as we mentioned in Fig. 1a, an approximately 6.5-μm-thick liquid layer remained between the polymer and the bottom electrode in a steplike structure (Fig. 4b). The profile of the edge of the liquid lens was measured with a three-dimensional laser microscope (VK-8700, Keyence Co.). The large peak presented in the Fig. 4b at approximately 0.4 mm is a scattered reflection occurring at the boundary between the CYTOP layer and liquid.

Fig. 4
figure 4

Deformation characteristics of the varifocal liquid lens. a Photographs of the varifocal liquid lens to which voltages ranging from 0 to 150 V were applied. b Comparison of the deformation profiles of the varifocal liquid lens. c Comparison of the applied voltage and the lens radius. d Comparison of the capacitance and the lens radius

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When we investigated the relationship between the applied voltage V act [V] and the radius of the spherical area of the liquid lens R lens [mm], we confirmed that R lens became smaller when the magnitude of V act was increased from 0 to 150 V (Fig. 4c). The value of R lens was measured from the images, such as Fig. 4a. At the same time, a monotonically increasing relationship between R lens and the capacitance change ΔC lens of the liquid lens was obtained (Fig. 4d). This result confirmed that the capacitance of the liquid lens changes due to deformation, thus demonstrating that the shape of the liquid lens can be detected from its capacitance change. To confirm this deformation model of the varifocal liquid lens, we measured the relationship between the shape of the lens and its capacitance and compared this result with the result calculated from the theoretical model. The comparisons are shown in Fig. 5. We calculated the theoretical capacitance of the lens by taking the sum of the spherical center part of the liquid lens and the parallel plate capacitors around it as shown in Fig. 1a. The capacitance of the spherical part was calculated according to [17]. If we do not consider the steplike structure as a simple model, which we proposed in [10], the theoretical capacitance of the lens becomes ten times greater than the measured result as shown in Fig. 5a. However, when we consider the height of the steplike structure, the order of the theoretical capacitance of the lens matches the measured capacitance (Fig. 5b). From this result, we confirmed that our detailed model, which considers the influence of the steplike structure, agrees with the actual deformation of the liquid lens.

Fig. 5
figure 5

Comparison of measured capacitance change and the theoretical results. a Comparison of 3 cases: measured result, theoretical results of the model that considers the steplike structure, and the theoretical result of the model that does not consider the steplike structure, b enlarged view inside the dotted area

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The relationship between the focal length of the liquid lens and the applied voltage is shown in Fig. 6. The applied voltage was increased from 0 to 150 V in steps of 15 V and returned to 0 V with the same step size. One-minute intervals elapsed between each step. As shown in Fig. 6, hysteresis existed in the relationship between the applied voltage and the focal length of the liquid lens. For example, in the range of 45–120 V, the focal length of the liquid lens when the applied voltage was increased was shorter than the focal length obtained when the applied voltage was decreased. This hysteresis resulted from the deformation lag of the liquid lens.

Fig. 6
figure 6

Relationship between the applied voltage and focal length of the varifocal liquid lens

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However, the relationship between the capacitance change of the liquid lens and its focal length is shown in Fig. 7. To measure the focal length, a 150-Vp–p, 0.1-Hz sinusoidal wave was applied to actuate the liquid lens. The result of this wave application demonstrated that the focal length of the liquid lens exhibited a good linear relationship with its capacitance change (correlation factor was R 2 = 0.996), and no hysteresis existed between them. From this result, the relationship between the capacitance change ΔC lens [pF] and the focal length F lens [mm] of the liquid lens was determined as

Fig. 7
figure 7

Relationship between the capacitance change and focal length of the varifocal liquid lens

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$$F_{\text{lens}} = K_{\text{lens}} \cdot \Updelta C_{\text{lens}} + F_{0} = - 1.66 \cdot \Updelta C_{\text{lens}} + 145.06$$
(3)

In this equation, K lens [mm/pF] is the coefficient that indicates the relationship between the capacitance change ΔC lens [pF], and the F 0 [mm] is the intercept of the equation. Using this relationship, the focal length of the liquid lens can be calculated from its capacitance change.

3 Distance measurement using a confocal distance sensor

3.1 Principle of the confocal distance sensor using the varifocal liquid lens

To evaluate our varifocal liquid lens and the method used to measure its focal length, we developed a confocal distance sensor using the fabricated liquid lens, as shown in Fig. 8. The laser was collimated with a glass lens and irradiated a target object through a polarization beam splitter, a quarter-wave plate, and the liquid lens. The light reflected on the surface of the object was polarized at the quarter-wave plate and entered the pair of pinholes and the photodiode via reflection at the beam splitter. The laser source, pair of pinholes, and photodiode were arranged at the same optical length from the liquid lens.

Fig. 8
figure 8

Principle applied to measure distance using our proposed confocal optical system. a When the laser is focused on the target. b When the laser is not focused on the target

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As shown in Fig. 8a, when the laser spot is focused onto the object’s surface, the reflected light returns along the same light path as that of the incident light, and the reflected light enters the photodiode through the pinhole. However, if the laser spot is not focused onto the object’s surface, the reflected light spreads at the pinhole surface, and the optical intensity of the reflected light that enters the photodiode becomes small (Fig. 8b). From these characteristics, the distance from the confocal optical system to the target object can be detected by measuring the focal position of the light when its reflection intensity peaks.

The proposed confocal optical system was used to scan the laser spot by applying a low-frequency, high-voltage sinusoidal wave to the varifocal liquid lens. The capacitance change of the actuated liquid lens and the intensity of the reflected light that enters the photodiode are measured simultaneously. The distance from the system to the object is calculated from the capacitance of the lens when the reflected light intensity is at its peak value. A photograph of the fabricated confocal optical system is shown in Fig. 9.

Fig. 9
figure 9

A photograph of the confocal optical system

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The laser source used in the fabricated confocal distance sensor had a diameter spot size of 4 μm, wavelength of 488 nm, and light intensity of 5 mW (HPU50101-PFS, Furukawa Electric Co.). The diameter of the pinhole in front of the photodiode was 100 μm. The spectral response range of the photodiode was 320–1,000 nm (S6931-01, Hamamatsu Photonics K.K.).

3.2 Evaluation of distance-measuring accuracy

In the experiment, we used an aluminum mirror as the target object and measured the distance to it. The distance to the mirror was set at 94–140 mm from the tip of the confocal distance sensor. A 150-Vp–p, 0.1-Hz sinusoidal wave was applied as the actuation signal V act to continuously actuate the liquid lens. Because the magnitude of the electrostatic force does not depend on the polarity of the applied voltage, the liquid lens deformed at twice the frequency of the applied V act. Thus, the liquid lens was deformed at 0.2 Hz in this experiment. To measure the capacitance of the liquid lens, the circuit shown in Fig. 1b was used.

An example of the capacitance change of the dynamically deformed liquid lens and the response of the photodiode to the laser reflection intensity is shown in Fig. 10. In the case of Fig. 10, the mirror was set in 110 mm far from the sensor. The distance to the target object was calculated by inserting the capacitance change of the lens into Eq. 3. To remove the power supply noise from the output, we used fast Fourier transform (FFT) analysis and removed signals higher than 2.5 Hz from the output. According to the peak of the laser reflection intensity in Fig. 10, the laser focused on the object surface twice in one deformation cycle of the liquid lens. This characteristic enabled the distance to be measured at a frequency of 0.4 Hz, which is 4 times higher than the frequency of the applied actuation signal V act.

Fig. 10
figure 10

Comparison of the capacitance change of the liquid lens and the intensity of reflected light

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A comparison between the actual positions of the target object set in 94–140 mm from the fabricated sensor and the measured distance is shown in Fig. 11. Based on this result, we confirmed that the distance of the object from the point at which it is actually located can be measured with an average error of 0.83 mm. The average standard deviation over 6 trials was 0.77 mm. According to this result, we confirmed that the proposed varifocal liquid lens and the proposed method of measuring its focal length based on the capacitance change can be used to realize optical systems that require highly accurate focal position control, such as confocal distance sensors.

Fig. 11
figure 11

Distance measurement using the fabricated confocal optical system with the varifocal liquid lens

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4 Conclusions

A varifocal liquid lens with a focal length that can be measured and adjusted without using extra sensors or other mechanisms was fabricated. The varifocal liquid lens had a diameter of 10 mm and a central thickness of 120 μm, and its focal length could be varied from 151 to 82 mm by applying a voltage ranging from 0 to 150 V. The capacitance change of this liquid lens exhibited a linear relationship with the focal length (R 2 = 0.996). Using this relationship and the capacitance change, the focal length of the liquid lens can be calculated. We created a confocal distance sensor using the varifocal liquid lens to scan the laser spot. Using this sensor, the distance to the target object, which was set between 94 and 140 mm from the system, can be measured to within an average error of 0.83 mm by applying a 150-Vp–p, 0.1-Hz sinusoidal wave to the liquid lens to dynamically actuate it. Thus, we confirmed that our varifocal liquid lens can be adjusted and that its focal length can be measured using only the lens itself. We were able to realize an optical system that requires highly accurate adjustment of the focal position, such as a confocal distance sensor.