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Abstract
The function of electrowetting liquid lenses is expanding beyond tunable focal length lensing. Recently, single and multi-mode oscillations on the meniscus profile of two non-miscible liquids have been used for optical phase modulation and fast focal length sweeping. To achieve a user-defined phase modulation, a prediction model of oscillation patterns and amplitudes is needed. We present digital holographic interferometry (DHI) measurements of oscillation patterns and amplitudes on a 5.8mm aperture lens up to 160 Hz, including frequency responses from 26–100 Hz. We discuss using Bessel function and Legendre polynomial models for oscillations on a conical frustum shaped electrowetting lens.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electrowetting lenses offer fast focal length tuning in a small package. The focal length is tuned by an electric potential change [1–4], which alters the meniscus profile between two non-miscible liquids of dissimilar refractive indices. The phase change is not polarization dependent, and consumes relatively low amounts of power [5,6]. Commercial electrowetting lenses provide fast switching between stable focal lengths with low hysteresis [4].
Uses for the often undesirable liquid lens oscillations have recently been proposed and demonstrated. There are reports on focal length sweeping using resonant oscillations for biomedical imaging applications [7,8]. There are also studies of optical phase modulation [9,10] and time-dependent image encryption [11] using combined oscillations from multiple driving frequencies or pulses. Combined oscillations provide more degrees of freedom than a stable profile, offering the ability to create a user-defined lens profile.
Wavefront shaping and modulation is possible using a near static lens shape, but requires a more complicated design. Multiple electrodes around the outside of a lens can be used for asymmetric wavefront correction [12]. Alternatively, multiple lenses positioned in an array with individually controlled electrodes becomes a spatially discrete tunable wavefront correction device [13]. With either of these design concepts, degrees of freedom are added at the cost of multiple electrodes and complexity. We focus on meniscus profile oscillations from one outer electrode, because multiple mode oscillations in a superposition can achieve axis-symmetric profile shaping with minimal design complexity.
In order to oscillate commercial or custom liquid lenses to create user defined shapes, oscillation amplitudes and patterns for given forcing functions must be determined from imaging experiments. The liquid contact angle can be estimated from measurements using incoherent, broadband illumination from the bottom of the lens [14]. However, this gives no other measurements, and would be difficult to implement for conically shaped lenses. Though imaging of liquid lens meniscus profiles is often done through the lens side-wall [2, 5, 14], commercially available lenses do not have optically transparent side-walls. When building a custom liquid lens, it is more convenient and cost-effective to deposit optically opaque side electrodes, such as metals. Therefore, imaging measurements through the top and bottom of the lens (center axis) and not through the side-wall is ideal.
In this paper, we present experimental forcing of electrowetting lens oscillations and the use of digital holographic interferometry (DHI) through the center axis of the lens to estimate the shape and amplitude of the oscillations. DHI is an interference technique that measures geometric changes of an object within 1/50 of the optical wavelength λ [15, 16]. This potential for high resolution makes DHI suitable for measuring nanometer scale geometric changes on a meniscus profile. Details of how to use reference beam DHI for imaging liquid lenses is presented for the first time. These measurements are compared to fitted profile shapes from a time-periodic Bessel function model. Some of the challenges in taking accurate measurements are noted. The limitations of a Bessel-function model are discussed, as well as a potential replacement model for conical frustum shaped lens oscillation modes.
2. Methods
A simple Bessel function based model for electrowetting lens meniscus profile oscillations is given in this section. Though purely cylindrical designs have been studied [2], conical frustum shaped lenses have prevailed as the main option commercially [4], and so we use them in this study. The modeling of the lens is followed by a brief description of reference beam DHI and the custom interferometer that was used for the presented measurements.
2.1. Modeling oscillations
The shape of the meniscus profile in the electrowetting lens, when forced with direct current (DC) or high frequency (>1kHz) bipolar square waves, can be predicted within a saturation and threshold range. The forced contact angle θF between the polar liquid and the side-wall can be modeled with the Young-Lippmann equation as [1,17]
where ∊ and d are the dielectric permittivity and thickness of the side-wall, γLL is the liquid-liquid surface tension, and θE is the equilibrium angle of the polar liquid. A flat profile at the liquid interface can be achieved with a non-zero DC voltage UB, as shown in Fig. 1.
Fig. 1 The cross-section of a conical frustum electrowetting lens. The first liquid (gray) is non-polar and has a refractive index n1, while the second liquid (blue) is polar and has a refractive index n2. The electric potential Ũ = UB + U0 cos(2πfft) is used to force a flat interface with a DC offset (in this case UB = 48V) and oscillations with a harmonic function.
When the contact angle changes periodically, the meniscus profile oscillates. Experimental observations have verified that at low amplitudes and frequencies, the shape of these oscillations can be modeled with periodically changing Bessel functions [9,10]. With this model, displacements from a flat surface of radius R, forced by electric potential Ũ = UB + U0 cos(2πfft), are modeled as
where r is radial displacement from the center axis (see Fig. 1), t is time, and c is the wave speed on the meniscus. In this model, the amplitude P depends on the forcing amplitude and frequency [9, 10]. The first derivative of the displacement at the contact line between the two fluids and the solid boundary is related to a small angle αm from the flat profile plane with ((∂η/∂r)|r=R = tan(αm) [9]. The geometric definition of the angle αm is shown in the zoomed left side of Fig. 1. Though Eq. (1) shows a relationship , we expect a linear relationship between forcing amplitude and oscillation amplitude P ∼ (UB + U0), as is the case for open surface liquid sloshing [18].In dynamic situations, the contact angle is time and speed dependent and influenced by dissipating processes and flow fields in the vicinity of the contact line [14,17]. Accurate modeling of the dynamic contact angle requires empirically found coefficients for a friction term [14,19] in addition to the terms in Eq. (1). Though it has been found that the friction term significantly influences the dynamics of an impulse response [20], the friction term plays a less significant role in the shape of standing waves from harmonic voltage inputs. We therefore do not go into depth on the modeling of the dynamic contact line here.
When using two liquids that are density matched, the orientation of the lens and external vibrations do not have a noticeable affect on the meniscus profile. This has been verified in experimental studies with lenses of similar make to the one used in this study [1,3,9,10]. With matched liquid densities, the Bond number is very small and displacements from gravitational disturbances are negligible.
2.2. Digital holographic interferometry
Digital holographic holography (DHI) using a reference beam allows for reconstruction of a complex wavefield [15] at discrete times. A series of intensity interferograms recorded by a camera can be used to extract measurements of small displacements. We use an off-axis reference beam to capture the dynamic nature of the phase change from oscillations. A diagram of the custom interferometer that was used for this study is shown in Fig. 2.
Fig. 2 A custom interferometer for digital holographic interferometry (DHI) measurements. Red lines symbolize the coherent beam path, which separates at BS1 and combines incident upon the CMOS sensor at a relative angle α between the paths of the two beams ER and EO.
The intensity I from both the reference beam ER and objective beam EO at the plane of the complementary metal-oxide-semiconductor (CMOS) sensor (x, y) can be expressed as
The first term, the intensity of the reference beam, can be measured separately while blocking the objective beam. With the assumption that the reference beam is not changing significantly within the measurement time, this term can be subtracted digitally.To extract the complex wavefield of the objective beam at the plane of the lens meniscus profile, we multiply by the conjugate of the estimated reference beam wavefield at the sensor, giving
then use a convolution approach to numerically compute the complex field of ER* · (I − |ER|2) at a propagation distance equal to the distance from the lens meniscus profile to the sensor. This gives three distinct complex images: the reduced image from the ER*|EO|2 term, and the two conjugate images that contain phase information. A region of interest around one of the twin images gives the complex objective beam image Γ(l, w, t), where l and w are pixel values for a grid L × L with Δl = 2R/L. The difference in the phase of Γ(l, w, t) from frame to frame, Δϕn = Arg{Γ(l, w, tn)} − Arg{Γ(l, w, tn−1)} is a measurement of how the meniscus profile is changing the phase of the objective beam over discrete time changes Δt = tn − tn−1. Within a scalar diffraction approximation, the phase difference Δϕn at a specific ff and U0 can be equated to where k is the wavenumber.If there are a sufficient number of frames per second, the amplitude of the oscillations can be estimated by measuring the phase difference at the center (l = w = L/2). At time step n, this is
which was used to determine the amplitude, frequency, and phase of the oscillations.3. Results and discussion
The custom interferometer shown in Fig. 2 was used to measure oscillations in a 5.8 mm aperture Varioptic (now Corning [21]) electrowetting lens with a shape that resembles Fig. 1. The two liquids in the lens had refractive indices of n2 = 1.3984, and n1 = 1.5196. A He-Ne continuous wave laser with a wavelength λ = 632.8 nm was split into an objective and reference beam. The angle between these beams when incident on the CMOS sensor was α ≈ 8.90 × 10−3 radians, while the distance from the meniscus profile to the CMOS sensor was 613 mm. A high-speed Phantom v711 CMOS camera with 20μm pixel width was used to record at 13000 fps and a 3 μs exposure time.
The lens was forced to an approximately flat profile with a 48V amplitude 1070 Hz square wave signal. Driving the lens with a direct current (DC) voltage for an extended period of time can result in catastrophic failure. Low frequency (ff = 20–200Hz) harmonic signals U0exp (i2πfft) were added to the square wave signal. While this means that the time-spectrum of the signal had higher frequency components from the square wave, it was observed that that these components did not excite oscillations with frequencies near that of the harmonic signals.
The experimentally measured interferograms contoured periodically with a frequency that matched that of the harmonic forcing signal. Figures 3(a) and Fig. 3(b) show a measured interferogram for what was found to be the first resonant frequency ff = 38 Hz and Fig. 3(c) shows the intensity of the three images after numerical propagation. The phase change around the lens region Δϕn is shown in Fig. 3(d).
Fig. 3 (a,b) Measured intensity interferogram while ff = 38Hz and U0 = 3.0V. (c) The intensity after numerical propagation to the image plane. (d) The phase difference (radians) between the current and the previous frame (not shown) in the region of interest.
The amplitudes of oscillations were estimated using a non-linear least squares (NLLS) fit. Taking advantage of Eq. (8), the fitting function was Ãsin(B̃t + ϕ̃). Figure 4(b) shows the least-squares estimated function plotted with measurements at the center (l = w = L/2) when ff = 38 Hz and U0 = 3.0V. The adjusted r2 value for this example was 0.95, and the mean adjusted r2 value for the plotted measurements in Fig. 4(a) was 0.92. The red markers in Fig. 4(a) show the amplitudes P when equating à = −2kP(n2 − n1) sin(πffΔt) from ff = 20–200Hz. The mean of the phase difference was subtracted for each frame for robustness. The mean, however, is non-zero for 2πfftn−1 + πff Δt ≠ 2πm, for some integer m. Therefore, this subtracts both signal and any offset from noise. If the signal mean was added back to the amplitude, Fig. 4(a) would maintain the same shape, but with slight amplification.
Fig. 4 (a) The amplitude of meniscus oscillations for different forcing frequencies using the summed profile measurements (black markers) and the NLLS fit to peak amplitude measurements at each frame (red markers). (b) The amplitude of the center point after subtracting the mean over three full oscillation periods (3τ) when ff = 38 Hz (blue markers), with the non-linear least squares estimation shown as a red continuous line. (c) The mean phase difference subtracted for each amplitude in (b).
The discrete time spectrum of the mean of the phase difference contained peaks at the forcing frequency and two higher frequencies. A plot of the mean phase difference for the same measurements that produced the amplitude plot in Fig. 4(b) is shown in Fig. 4(c). This amplitude offset is partially due to the meniscus profile not being perfectly stable, even without the harmonic forcing U0 cos(2πfft). The 1070 Hz bipolar square wave can hold the liquid lens at a certain contact angle in a near equilibrium state, but slight movement persists. Measurements of the profile without the U0 cos(2πfft) forcing revealed periodic shifting of the profile. The spectrum of this signal has peaks at the first resonant frequency, 429 and 1468 Hz, with the largest peak at 429 Hz. The maximum amplitude was 5.15μm, and the measured variance was 4.0 μm.
To measure the amplitude of the oscillations, the phase difference can be summed over multiple frames. Using estimated peak signals, such as the one in Fig. 4(b), the phase of the oscillations was determined. Starting with the frame associated with a flat profile, phase changes were summed until the frame associated with a maximum amplitude profile. This was done over two oscillation periods for averaging.
Noise in the measurements was mostly zero-mean, and canceled in the summed profiles. The summed offsets h̃ were subtracted from the final profiles using the formula
where P̃p is the estimated amplitude of the Bessel function shape (with the offset), r̃c is the ratio P̃p/P̃a, where P̃a is the measured anti-node value, and rc is the expected ratio. The expected ratio rc does not change with frequency, and so the offset h̃ can be estimated with only P̃p and P̃a. Figure 5 shows the cross-section of a summed profile before and after the offset was subtracted.Fig. 5 The cross-section of the measured summed profile for ff = 38Hz before (black) and after (red) subtracting the offset h̃.
The resulting summed profiles were used to compile an amplitude frequency response for 26–100 Hz. The amplitude of summed profiles after the offset was subtracted are shown in Fig. 4(a). The amplitude was found to increase linearly with voltage, as expected. The system is heavily damped, making resonant peaks less pronounced. The response from U0 = 1.8V has a peak at f1 = 38Hz. Using the membrane model proposed in [9,10], the wave speed can be calculated to be c = 2πRf1/j11 = 24 ± 1 cm/s, which can be compared to the calculated c = 26 ± 6 cm/s in [9] for a smaller diameter lens, made by the same company.
A two-dimensional plot of the summed profile phase for ff = 38Hz and ff = 160Hz is shown in Fig. 6. The cross-sections of the measured profiles are compared with the corresponding Bessel function model from Eq. (2) in Fig. 6(b) and 6(d). The phase measurements are limited to a region r < 0.75R due to the 5.8mm aperture. For the ff = 38Hz cross-sections, the root mean squared (RMS) difference was 25nm and 13nm for U0 = 1.8V and U0 = 3.0V when r < 0.50R, and 47nm and 34nm when including all data points r < 0.75R.
Fig. 6 The measured amplitude profile for (a) ff = 38Hz and (c) ff = 160Hz. The cross-section of the measured and modeled (Eq. (2)) profiles for (b) ff = 38Hz and (d) ff = 160Hz with forcing amplitudes U0 = 1.8, 3.0, 4.0, 8.0V.
The Bessel function model did not match the experimental results consistently. For frequencies 26 – 100Hz, The wave speed followed a frequency dependent model c̃ = ac2π/(j11(1 + γ/ff)), for a constant γ, but diverged for higher frequencies. It can be seen in Fig. 6(b) that the profiles compare well with the model at smaller radial values (r ≤ R/2 for ff = 38Hz), but diverge at greater radial values. Also noticeable from Fig. 6(b) and 6(d) is how this divergence increases with frequency and forcing amplitude U0. The three phase nodes in Fig. 6(d) alternate in magnitude, going from high (center) to low (2nd) to high again (3rd), in contrast with a Bessel function, which has nodes gradually decrease in value from the center. The simple Bessel function model, Eq. (2), is derived from the wave equation for cylindrical membranes. The velocity potential for this model at the surface for the nth mode is , for eigenfunction , where is a Bessel function root [18]. When the Laplace equation is solved for a velocity potential in a conical frustum geometry, the potential at mode n is [18,22]
where θ and rs are defined by the coordinate system in Fig. 7(a), λn is the eigenvalue, An is the coefficient for the nth mode, and Rs ≫ h1, where h1 is the height or thickness of the first (bottom) liquid. This solution is similar to one found for oscillating spherical drops [23], with the notable difference in the (rs/Rs)−λn−1 term, which is neglected for hemisphere or full cone geometries as rs approaches zero in the domain.Fig. 7 (a) A spherical coordinate system for the lens geometry can be used. A plot of the velocity potential at the meniscus for Legendre polynomial (conical shape solution) and Bessel function models (cylindrical shape solution) when the (b) fourth eigenvalue and (c) seventh eigenvalue of the Bessel function equation was used.
The Legendre polynomial model was compared with the Bessel function model for the fourth and seventh modes along the meniscus profile. The parameters An and λn of the Legendre polynomial model were tuned to match the Bessel function model around the center r = 0. The divergence of these two models resembles the divergence of the Bessel function displacement model with the measurements, as shown in Fig. 6(b) and 6(d). The Legendre polynomial model plotted in Fig. 7(c) allows for the second and third nodes from the center node to decrease in magnitude, then increase, which matches qualitatively with the measurements shown in Fig. 6(d).
4. Conclusions
Digital holographic interferometry (DHI) measurements of forced electrowetting lens meniscus profile oscillations have been presented. The amplitude of the oscillations can be effectively measured by summing the phase changes at each frame over 1/4 of the oscillation period. Any offset of phase can be subtracted by using the ratio of the measured meniscus profile center node to the first anti-node. The frequency response shows that the system acts like a heavily damped second order system. The oscillation amplitudes increase linearly with forcing amplitude, and the oscillation frequency corresponds to the forcing frequency. This offers an advantage for dynamic focusing applications, such as focus stacking. If one looks at the second order term of an expansion about the r = 0 axis, the maximum optical power PO of this central paraboloid is related linearly to the forcing amplitude for a given frequency PO ∼ U0.
A Bessel function model, from the wave equation in cylindrical geometry, was fitted to the measurements. For a 5.8 mm aperture conical frustum shaped commercial lens, the first resonance frequency was measured to be f1 = 38Hz, giving a wave speed of c = 24 ± 1 cm/s. The wave speed depended on frequency, however, making it only predictable for frequencies ff ≠ f1 with a non-linear equation. In the example measurements shown for ff = 38Hz and U0 = 3.0V, the RMS difference was λ/50 when r < R/2, and λ/14 for all data points. The difference increased significantly as the radial distance approached the contact line.
This can be explained by the conical frustum shape of the lens. The Bessel function model is for a cylindrical lens. A Legendre polynomial velocity potential for a conical frustum shape was plotted against a Bessel function solution for cylindrical shapes. When comparing the velocity potential of the two models at the meniscus, they tend to diverge in similar fashion to the measurements. An analytic solution to accurately model conical frustum shape liquid lens oscillations is currently being developed. A limitation of multi-frequency or pulsed forcing of user-defined meniscus profiles for wavefront shaping is the accuracy of the model basis function. A more accurate basis function will enable less residuals. This limitation also applies to a multi-electrode system with additional degrees of freedom, that would potentially be modeled with a Bessel function or Legendre function of higher order.
Acknowledgments
The authors thank Stephen Phillips for his assistance with some of the measurements, and Dr. Matthias Strauch of the Delft University of Technology for relevant discussions and help with the lens driving signal.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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