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Abstract
We present a highly compact and fast varifocal lens with aspherical tunability based on an active piezo-glass-piezo sandwich membrane. Using an optimized geometry, improved fabrication and compliant elastomer structures together with an index-matched optical fluid, we achieved an outer diameter of just 9 mm (10 mm packaged) for a clear aperture of 7.6 mm. The range of the focal power was -7 m−1 to +6 m−1, with a wavefront error around 100 nm and a response time between 0.1 and 0.15 ms.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Varifocal lenses [1] enable focusing or scanning of optical systems without moving lenses, hence reducing complexity and power consumption and improving the response time. In general, desirable properties are a short response time, compact size for a given aperture, low power consumption, high quality and, ideally, aspherical tunability. In this paper, we present the development of a lens that has the potential to combine these properties.
The most common approach to tunable lenses uses a membrane that covers a fluid volume and forms the lens surface. Traditionally, the membrane is displaced by pumping a fluid into the volume under the membrane, using either an integrated annular pump around the membrane [2–4] or an external pump [5,6]. Alternatively, the membrane may be deformed directly using a fluid [7], gel [8] or elastomer [9] as a mechanically passive refractive medium. The other major approach is electrowetting, where the interface of two gravity-matched immiscible fluids forms the lens surface and is deformed through electrostatic forces [10]. Liquid crystals are used less commonly to create a lens effect [11,12]. These approaches have in common a response time of $\def\upmu{\unicode[Times]{x00B5}}2\,\mathrm {ms}$ or more. Going beyond this limit, tunable acoustic gradient (TAG) lenses [13,14] use the pressure field of an acoustic resonance in a cylindrical lens body to create an inhomogeneous refractive index and hence a lens effect. While they operate at several $100\,\mathrm {kHz}$, they can only be used in pulsed applications and have hence limited use. With the exception of [7,9] (which we discuss below) and [6] (and other eyeglasses, which are not the focus of this research), these varifocal lenses have only half or less of the device diameter optically usable.
In [7], we introduced a novel active piezo-glass-piezo sandwich membrane where two piezo rings deform a thin glass membrane as shown in Fig. 1. This membrane is mounted on top of a fluid chamber which, in contrast to conventional fluid-membrane lenses that actively pump a fluid into the lens, only plays a passive role as a refractive medium. The reduced moving mass in combination with the stiff membrane and the fast piezo actuator gives a response time of 2 to $3\,\mathrm {ms}$ and the compact actuator and absence of an external or integrated pump results in an outer diameter of only 19.4 mm for 12 mm free aperture. We developed a simulation model that takes into account the mechanical nonlinearities and phenomenologically includes the piezoelectric creep and hysteresis in [15], where we systematically studied the behavior of the active membrane. The main limitation for the size was the elastic support ring that was needed to compensate the fluid volume displaced by the deformation of the glass membrane.
Fig. 1. Concept of the first prototypes: Active membrane (top left), fluidic lens (left) and elastomer lens (right).
To circumvent this limitation, we used in [9] an elastomer as a refractive medium instead of the fluid (Fig. 1, right) and produced a device with 6.25 mm clear aperture and 7.9 mm diameter (8.5 mm including contact pads, 9 mm packaged). While the system worked in principle, a PDMS elastomer turned out to be too stiff with a focal power range less than 3 $\textrm {m}^{-1}$ and the step response was affected by strong ringing extending to approximately $10\,\mathrm {ms}$. A polyurethane elastomer, in contrast, was much softer and provided a focal power of $\pm$10 $\textrm {m}^{-1}$. It showed, however, a strong birefringence and a very slow response of several $100\,\mathrm {ms}$ due to its viscoelastic properties.
In this paper, we hence revisit the fluid-based principle and present the optimization for a large clear aperture, large focal power and high surface quality of the lens, keeping an outer diameter of 9.0 mm. A short response time will also be an outcome of the small dimensions. We briefly presented a first prototype in [16] which indicated a high speed but did not have satisfactory quality. In the present paper, we present the fully optimized lens and the details of the design, optimization and fabrication, and a detailed characterization. We will first present the optimization of the design in section 2.1, the materials in section 2.2 and the fabrication in section 2.3. In section 3, we will present a mechanical characterization of the lens surface in a bidirectional (concave and convex) and unidirectional (concave only) configuration, first the focal power and aspherical behavior and then the dynamic response.
2. Design and fabrication
As in [7], the lens surface is created by a thin glass membrane that is sandwiched between two piezo rings with surface electrodes and out-of-plane polarization. These two rings then control the two boundary conditions of the glass membrane and give rise to buckling and bending deformations as we illustrate in Fig. 2 and as it was described in [17]:
- • The mean strain of both (assumed to be narrow) piezo sheets with radius $R$,$$\Delta R \ = \ R\, d_{31} {\textstyle \frac{1}{2}}\left(E_{\mathrm{low}} +E_{\mathrm{up}}\right)$$controls the Dirichlet boundary condition of the membrane. Keeping in mind that the transverse piezo coefficient $d_{31}$ is negative, the mean electric field of the sheets in the direction of the polarization, $E_{low} + E_{up} > 0$, will compress the membrane and hence cause a buckling deformation while negative electric fields will stretch the membrane and pull it flat. In a very crude estimate in which we assume a spherical deformation and ignore the back-reaction forces of the glass membrane and the bending moment of the piezo rings, the curvature of the membrane becomes
- • The difference of the strains, in contrast, will cause a bending deformation. Ignoring forces such as the mechanical back-reaction of the glass membrane and assuming the piezo rings to deform in an annular section of a sphere with some separation $h$ of the neutral planes of the piezo sheets, we can straightforwardly estimate the von Neumann boundary condition, In an assumed spherical deformation, this gives a curvature radius of
In combination, these boundary conditions then allow us to control both the curvature of the membrane at its center and the next subleading (quartic, $\propto r^4$) term in the deflection profile, i.e. the focal power and the spherical aberration.
As these considerations assume an idealized geometry and do not take into account forces and the mechanical back-reaction from the elastomer and from the internal pressure of the fluid, we used FEM simulations with COMSOL to optimize the design.
2.1 Simulation and optimization
To optimize the speed, we will keep the moving mass low and to minimize the footprint (maximize the clear aperture ratio), we will minimize the width of the piezo ring and choose a minimalistic design where we will only use a circular rigid base substrate (gray in the cross section in Fig. 3, left), a transparent elastomer ring (light red) and the sandwich membrane (gray/light blue). In general, we try to use the vertical direction and develop compliant structures. For manufacturing and reliability, we keep at least 500 $\upmu$m contact width at all glued surfaces, 180 $\upmu$m thickness of the elastomer membrane and 200 $\upmu$m wide contact pads of the piezo sheets. For this reason, we choose the three dimensional glue surfaces (green lines) between the elastomer and the membrane and bottom substrate. The outer wall of the elastomer is further assumed to be fixed by the packaging of the lens (red lines, red shading). As filling the lens involves punctuating this wall with a syringe needle, we require a minimum thickness of 250 $\upmu$m.
Fig. 3. Left: Cross section of the lens (final dimensions). Right: FEM simulation in the displaced (combined bending and buckling) state and illustration of the compliant principle of the elastomer structure. The red and blue arrows illustrate rotations and linear movements, respectively.
If the membrane deforms, e.g., in a convex profile, it will on the one hand rotate upwards at the edge and on the other hand draw volume from the fluid chamber. The vertical segment of the elastomer acts as a hinge and is constrained radially by the membrane and by the horizontal segment, resulting in a compliant motion indicated in Fig. 3, right. The drawn fluid is then displaced by the inward motion of the upper side wall and an overall vertical motion of the membrane.
To verify this mechanism and study the effect of the geometric dimensions shown in 3, we performed an FEM simulation using COMSOL multiphysics. As we are interested in the general mechanical effects, and to avoid non-trivial sequential simulations with modeled hysteresis as done in [15], we will only simulate the example of a mixed bending and buckling mode with $\left (E_{\mathrm {low}} , E_{\mathrm {up}}\right )\, =\, (1.2,\!-0.3) {{\frac {\mathrm {kV}}{\textrm {mm}}}}$ applied to the two piezo rings, where the lower value corresponds to 30% of the coercive field strength and the upper value was chosen to eliminate the risk of electrostatic breakdown. We had piezo sheets available in thicknesses of 120 and 150 $\upmu$m, for which we used a Young’s modulus of $127\,\mathrm {GPa}$ and the built-in PZT-5H with $d_{31}\, =\, - 274\, \mathrm {pm/V}$ with the voltages scaled by a factor of 1.5, corresponding to a typical $d_{31}\, =\, - 410\, \mathrm {pm/V}$ [18]. As the glass membranes (SCHOTT D263t eco) are commercially available in thicknesses of 30 $\upmu$m, 50 $\upmu$m and higher and as we observed a minimum aspect ratio of the membrane for buckling around 200 [15], we use the 30 $\upmu$m membranes. Using these conditions, we first simulated the sandwich membrane alone.
In Fig. 4, we see that the virtual focal power (assuming a refractive index of 1.48 for the fluid) has a maximum for a width of the piezo rings between 1 and 1.5 mm, increasing with increasing thicknesses. However, the spherical Zernike coefficient also increases with increasing piezo width. In fact, it is possible to change between elliptical and hyperbolical profiles by adjusting the PZT thickness. As too large spherical coefficients may not be useful in the overall imaging system, we introduced a usable aperture radius, defined as the minimum of either the clear aperture or the diameter over which the local curvature changes by less than a factor of 0.5, analytically extrapolated using the spherical coefficient. As a measure for the optical performance, we then used the resulting numerical aperture which has a maximum at a piezo width around 700 $\upmu$m and thickness of 120 $\upmu$m or higher. We chose the 120 $\upmu$m piezo material with 500 $\upmu$m wide rings as a compromise to maximize the usable aperture at a cost of a marginaly reduced numerical aperture.
Fig. 4. Simulated membrane deformation as a function of the width and thickness of the piezo rings. Left to right: Virtual focal power, spherical Zernike coefficient, numerical aperture and effective usable aperture. The red line indicates the minimum handleable width and the blue lines indicate the available thicknesses.
Using these parameters for the membrane, we simulated the effect of the fluid chamber assuming Smooth-On ClearFlex 50 polyurethane elastomer with a Young’s modulus of $2.5\, \mathrm {MPa}$ [17]. We assumed a nearly incompressible Poisson’s ratio of 0.49 between a rough value for thermoplastic polyurethane (0.48, [19]) and an accurate value for PDMS (0.495 to 0.497, [20]); varying this value from 0.45 to 0.499 has an effect of just 1% in the final results. We accounted for the nearly incompressible fluid by computing the volume change of the fluid chamber and applying a surface load to the inner surfaces (blue lines in Fig. 3) to keep the volume constant. To reduce the degrees of freedom to vary, we assumed equal thicknesses in the vertical and horizontal part of the fluid chamber and fixed the inner wall to the inner radius of the piezo ring, i.e., to the clear aperture. We see in Fig. 5 that the focal power decreases with increasing thickness and decreasing height of the fluid chamber, as this makes the fluid chamber stiffer. There is a small residual change compared to the pure membrane even at vanishing thickness, probably due to the bending moment of the remaining elastomer at the contact to the membrane. The spherical coefficient decreases with increasing stiffness of the fluid chamber, presumably due to the increasing counter pressure of the fluid acting on the membrane. It is even possible to change the behavior by more than 100% to an elliptic profile. The effective aperture remains unchanged. Hence, we chose the minimum elastomer thickness of 180 $\upmu$m and a height of the vertical segment of 770 $\upmu$m, where the focal power is suppressed only between 10 and 15% and the height does not become too large, in particular in the context of oblique incidence of light.
Fig. 5. Effect of the height of the vertical segment of the fluid chamber and the wall thickness of the elastomer on the membrane deformation. Left: Relative change in the virtual focal power. Right: Relative change in the spherical Zernike coefficient. The red line indicates the minimum reliable thickness and the star indicates the chosen values.
2.2 Lens fluid
To maximize the usable aperture, we hide the part of the lens chamber that reaches into the clear aperture and its glue to the membrane (also ClearFlex 50) by index-matching the fluid in the lens. The refractive index of the elastomer varies between $n_{D,20}\, = \, 1.4860$ and $n_{D,20}\, = \, 1.4864$ over different batches. We assume, however that this variance can be reduced by more reproducible processing and a post-curing or pre-aging schedule. As in our experience, including swelling tests, polyurethane is (only) fully compatible with aliphatic hydrocarbons, the candidates for the mixture shown in Fig. 6 (left) were viscous paraffin oil (typically $n\,=\, 1.476 \ldots 1.481$, in our case $n_{D,20}\,=\, 1.4778$ and $n_{D,20}\,=\, 1.4789$ for different batches), decalin ($n_{D,20}\,=\, 1.47927$ for our mixture of cis- and trans-decalin), perhydrofluorene (PHF, $n_{D,20}\,=\, 1.50181$) and perhydropyrene (PHPY, $n_{D,20}\,=\, 1.52268$) – see [21] as a reference. As paraffin has a significantly higher viscosity (approx. $200\,\mathrm {mPa}\,\mathrm {s}$ at 20℃ for our batches) than the pure compounds (e.g. $3.4\,\mathrm {mPa}\,\mathrm {s}$ for cis-decalin), mixtures based on decalin have a significantly lower viscosity, whereas we expect the mixture of PHPY and paraffin to have the highest viscosity because of the highest content of paraffin and the mixture of PHF and paraffin a lower viscosity. Cis-decalin has a vapor pressure of $240\,Pa$ [22], so decalin-based mixtures may be subject to evaporation by diffusion through the elastomer. We expect paraffin-based mixtures, however, to be stable.
Fig. 6. Left: Measured refractive index at 20℃ of (top to bottom) perhydropyrene, perhydrofluorene, two batches of polyurethane elastomer, one batch of paraffin oil and decalin (unknown mixture of cis and trans). Right: Difference between the refractive indices of polyurethane and the matched liquids. Note that both the polyurethane and paraffin oil come each from different batches, so the mixing ratio is only indicative.
In Fig. 6 (right), we see a good matching of the different fluids and the elastomer, with some mismatch of the order of $10^{-4}$ in the far blue and red. The different mixing ratios of the two paraffin-PHF mixtures result from the different refractive indices of the batches of paraffin and polyurethane. The PHPF-paraffin mixture has a better agreement than the mixture using PHF and the mixture of decalin with PHF performes better than paraffin with PHF. Hence, we assume that PHPF with decalin, which we did not try, might have the best match. The resulting low viscosity might, however, cause problems in high-speed operations as we will study in section 3.3. The system PHPY-decalin-paraffin is most economical and can be tuned to the desired viscosity.
2.3 Fabrication
We first laser-structured the glass membrane (30 $\upmu$m) and substrate (500 $\upmu$m) from SCHOTT D263t eco and the piezo rings including the 200 $\upmu$m wide contact pads from 120 $\upmu$m thick Johnson Matthey M1100 using a nanosecond UV laser. During the cleaning process of the piezo rings, we used a short dip in 25% $\mathrm {HNO}_3$ to remove residues of the ablated gold electrode and PZT material to avoid electrostatic breakdown along the edge of the material. We then successively glued the piezo rings to both sides of the the glass membrane using optical quality vacuum chucks of different sizes at a slightly elevated temperature around 35℃ to ensure a flat membrane and low thermal stress.
While we attempted to control the glue layer by dispensing it with a PDMS stamp, we also developed a two-step gluing process to prevent the glue from coating the contact pads before contacting: We used the Resoltech HTG 240 epoxy system, which uses cyclohex-1,2-ylenediamine as a curing agent that evaporates in air. Hence, the thin layer of glue that flows onto the contact pads but also the outer part of the desired glue layer will not cure. To contact the piezos, we again placed the membranes on vacuum chucks and used 50 $\upmu$m thick coated copper wires and bismuth-based low-temperature solder that can be soldered below the Curie temperature of the piezo material. The thin coating of uncured epoxy on the contact pads mixed with the solder flux and did not affect the soldering. Alternatively, to prevent deformations due to thermal stress, we also fabricated prototypes using silver/epoxy-based conductive glue (Panacol Elecolit 3025) with a special mount to fix the wires. This conductive glue actually uses the same curing agent as HTG 240. After contacting, we then exposed the membranes to a saturated atmosphere of the curing agent at approx. 35℃ to cure the remaining glue.
To complete the lens, we first cast the polyurethane elastomer structure from Smooth-On ClearFlex 50 in a two-piece mold and then glued the elastomer ring to the laser-structured groove in the rigid glass substrate, also using ClearFlex 50. Finally, we glued this structure onto the finished membrane using ClearFlex 50 and vacuum chucks.
During filling, we had to consider the extraction of air bubbles and the internal pressure. For this purpose, we mounted the lens vertically as shown in Fig. 7 (left), such that we could illuminate and observe it with a beam splitter through the substrate, fill it with a syringe from one side and extract air bubbles with a second syringe from the top. The top syringe was connected with a bypass valve to vacuum to actively extract air bubbles and adjust the internal pressure, i.e., the pre-displacement of the membrane. To control this process without optical artifacts due to the filling, we monitored the displacement of the PZT ring with a confocal sensor. We correlated this displacement to the membrane curvature by simulating the lens profile for a sequence of internal pressures We then iteratively measured the curvature with an optical profilometer, calculated the required shift of the membrane edge to level the curvature and re-adjusted the pressure/displacement accordingly. As the FEM simulation is not perfect, and the membrane also moves by inserting the syringes, we required typically 1-3 iterations to reach an offset below 0.2 $\textrm {m}^{-1}$. According to simulations, a lens with up to 0.77 $\textrm {m}^{-1}$ can still deflect in both directions using the maximum combined bending and buckling mode with (-0.3,1.0) ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ or using the pure bending mode with (-0.3,0.3) ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ for a lens with up to 0.23 $\textrm {m}^{-1}$, which is preferable.
Fig. 7. Left: Filling process. The inset shows the lens during the filling. Right: Images of an unpackaged and four packaged lenses, from left to right: Sealed package, open FR2 and polyurethane packages, bare lens, open cylindrical package; approximately using the same scale.
To close the lens, we finally glued the finished lens into the package, and connected the contact wires to a flexible PCB that acts as an FPC connector. Figure 7 (right) shows four different packages: A square-shape fiber-reinforced polyamide package sealed with a protective cover glass, circular mounts using rigid (FR2) or flexible (polyurethane) sidewalls and a cylindrical reinforced polyamide package for integration into an endoscopic high-resolution microscope. All packages have a side length or diameter of 10 mm, except for the cylindrical package with 10.7 mm because of the structural integration into the microscope. We also coated the outer edge of the composite membrane with ClearFlex 50 to reduce the risk of electrostatic breakdown; the inner edge is already protected by the meniscus formed by the glue on the glass membrane.
3. Characterization
The optical effect of the lens results from the surface profile, the pressure-induced change of the refractive index of the fluid and the optomechanical effect in the glass. To estimate the latter two effects, we use again the example of combined bending and buckling of section 2.1. The pressure in the fluid chamber was $200\,\mathrm {Pa}$, while the bulk modulus of paraffin oil is of the order of $1.6 \, \mathrm {GPa}$ [23]. Hence, the compression of the fluid is of the order of $1.2\times 10^{-7}$ and we can neglect its change of the refractive index, which furthermore is homogeneous and isotropic.
To estimate the optomechanical effect in the glass, we computed different stress invariants averaged over the thickness of the glass membrane at different radial positions. Because of the small thickness, we ignore higher order effects on oblique incident light from the inhomogeneity of the invariants. We also do not consider the effect of the glass substrate as it shows no deformation. The largest average of the absolute value of the first invariant of the stress tensor (worst assumption) was $8 \, \mathrm {MPa}$, while the largest modulus of the average linear stress vector (probably more relevant) was $6 \, \mathrm {MPa}$. With a stress optical coefficient of $3.4 \times 10^{-6} \mathrm {MPa}^{-1}$ [24], the change in the refractive index is at worst of the order of $\Delta \mathrm {n}\,\simeq \,3\times 10^{-5}$, resulting over the 30 $\upmu$m membrane in an optical path difference of less than $1\,\mathrm {nm}$. Hence, we conclude that the relevant optical information is entirely given by the surface profile.
To measure the surface profile, we scanned the clear aperture of the membrane on a grid of $46\, \times \, 46$ points on $7.5\, \times \, 7.5 \, \mathrm {mm}^2$ with an optical profilometer. For static measurements of the membrane alone, or the lens before filling, we used a chromatic confocal sensor with 4 $\upmu$m spot size and $17 \,\mathrm {nm}$ axial resolution to distinguish both membrane surfaces and for (quasi) static measurements of the filled lens, we used a sensor with 12 $\upmu$m spot size and $110 \,\mathrm {nm}$ axial resolution. In the quasistatic measurements, we used a periodic actuation with a rise rate of $1 \, \mathrm {kV/mm \, / s}$ to eliminate artifacts of the dynamics and obtain a good approximation to the static voltage-dependent lens profile. To measure the dynamic response, we used a laser-triangulation sensor with a measurement rate up to $380 \, \mathrm {kHz}$, but with a lower lateral resolution due to systematic errors of the triangulation principle on curved surfaces and a larger measurement spot of 25 $\upmu$m. It can also be affected by interference artifacts in the membrane.
3.1 Fabrication quality
To monitor the quality of the lenses during the fabrication process, we measured the surface profiles at different steps in the fabrication process. As the lenses can control their focus and aspherical behavior, we used the deviation from rotational symmetry as a measure for the quality, so we took the RMS deviation from an even $8^{th}$ order radial polynomial fit minus a possible tilt that is obtained by minimizing the discrete Integral
In table 1, we monitor the focal power offset, the RMS wavefront error, the astigmatism and the higher order wavefront error (after subtracting the astigmatism from the profiles) for two fabrication batches during the fabrication. From the wavefront error, one can, e.g., compute the Strehl ratio $S \, = \, e^{-(2\pi W_{RMS}/\lambda )^2}$[25]. We see that there was a significant improvement from the first batch to the second batch, even though the fabrication method remained the same. The main differences were a parallelization of the process and more careful cleaning and handling of the components and of the vacuum chucks. Judging from the distortion profiles and the radial scaling of the RMS distortion, we found that membranes with large surface errors were dominated by an astigmatism, i.e., by large defects, internal stress or particles. The highest quality membranes were dominated by higher order errors, which may result from the small-scale unevenness and the roughness of the piezo. This is on a similar level for all lenses as we see from the small standard deviation of the higher order wavefront error. There was also a radius-independent component of the order of 10 to 20 nm, which may result from the quality of the glass.
Table 1. Mean offset of the focal power, wavefront error $W_\mathrm {RMS}$, astigmatism $\left |a_2^{\pm 2}\right | \, := \, \sqrt {{a_2^{-2}}^2+{a_2^{2}}^2}$ and wavefront error of higher order aberrations (without astigmatism) $w_\mathrm {RMS}$ in the steps of the fabrication process. All values assume a refractive index of 1.48 in the lens. The values in parentheses give the standard deviation.
Contacting the piezo, re-curing it and gluing it to the lens chamber resulted in a small improvement of the RMS value, in particular for the most distorted membranes, which may result from the stabilization by the lens chamber, or from a stabilizing effect due to the new curvature offset. There seemed to be, however, no significant difference between the soldering process (batch 1) and conductive gluing (batch 2). The large focal power offset may be due to an internal pressure in the lens or due to a torque acting at the contact point of the lens chamber and the membrane. Both may naturally result from the deformation of the elastic chamber when the chamber is pressed onto the membrane and remains sucked to the membrane. Some pressure and movement was always necessary to ensure a good fit at the contact point.
Filling the lens in the iterative controlled process resulted in very flat lens surfaces, with offset focal powers typically well below 0.1 $\textrm {m}^{-1}$. After mounting the lens and connecting the contact wires, the offset however rose to 0.56 $\textrm {m}^{-1}$, which may result from the imperfect lens chambers. On the one hand, they have an irregular outer surface, so they may be distorted directly during mounting. For this reason, we introduced, on the other hand, a significant gap of around 100 $\upmu$m between the mount and the fluid chamber. Shrinkage and capillary effects may then cause a negative pre-deflection of the lens. There is also a risk of a negative capillary pressure acting on the oil through the syringe cuts from the filling process, such that small amounts of fluid might be drawn out of the lens. Still, the lenses filled with the controlled process had a much lower offset than the lenses filled without feedback. The large wavefront error of the lenses without uncontrolled filling likely results from the relatively short contact wires in the square packages, which may exert forces onto the membrane. In one lens, we attempted to re-pole the piezo sheets by applying a forward voltage of 1 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ for several minutes. The small decrease in the wavefront error indicates some effect due to inhomogeneous depolarization, e.g., during the laser structuring or during the soldering.
For illustration, we show the distortion profiles (after subtracting out the rotationally symmetric pre-deflection) of the best and worst lens that were fabricated with a controlled filling process in Fig. 8. Again, the worst lens is dominated by an astigmatism, roughly in the direction of the contact pads, whereas the best lens is dominated by higher order distortions (coma).
Fig. 8. Surface deformation (not taking into accound the rotationally symmetric pre-deflection) of the best and worst lens with controlled filling process.
All lenses from the first batch contacted by soldering worked reliably during the tests. Several of the lenses of the second batch contacted by conductive gluing, however, suffered electrostatic breakdown, typically near the outer edge but not always near the contact points. There is some possibility that the conductive glue affected the outer edge despite careful work under a microscope. The lenses of the second batch were, however, also fabricated from a new batch of piezo sheets, so there may also be a problem of the piezo material.
3.2 Quasistatic behavior
To determine the operating region of the focal power and the aspherical behavior, quantified as the Zernike coefficient $a_{4,0}$ on a radius of 3.2 mm, we measured the lens surface along different voltage trajectories shown in Fig. 9, left. To stay in the stable regime, we scanned the full voltage range separately in two parts – always keeping one voltage larger or equal to the other. Crossing between the half-planes in a buckled state, e.g., going from (1,0) ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ to (1,1) ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ and then to (0,1) ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ could provide a larger operating region by adding metastable states, however at the expense of operational stability. In Fig. 9 right, we show the results of such a scan over all voltage combinations at the example of a highly pre-deflected lens. To obtain a reproducible result, we measured along paths of increasing voltage (solid narrow arrows) and to avoid artifacts from long-term drifts we referenced the measurement to a static measurement at (0,0) ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$. As we can see by moving along lines of constant focal power (black contours), we can tune the focal power and spherical coefficient (colors) independently within some operating region.
Fig. 9. Left: Different voltage trajectories for characterization. The thin black and grey arrows indicate a full scan, separately for both semi-planes, the data was used only from the solid segments. The thick black and grey arrows (dashed along the diagonal) represent the voltage sequence that outlines the operating region and the blue and red (dashed) arrows indicate the pure bending and buckling modes. Right: Corresponding focal power (contours) and spherical Zernike coefficient (colors) for a highly pre-displaced lens.
To save measurement time and better illustrate the operating region, we measured along voltage trajectories that outline the parameter space, the thick black and gray arrows in Fig. 9, left: $(-0.3,\!- 0.3) \, \rightarrow \, (-0.3,\!1.0) \, \rightarrow \, (1.0,\!1.0) \, \rightarrow \, (-0.3,\!- 0.3) \, \rightarrow \, (1.0,\!-0.3) \, \rightarrow \, (1.0,\!1.0) \, \rightarrow \, (-0.3,\!- 0.3) \mathrm {kV/mm}$. In Fig. 10, we compare on the top left three representative lenses with different pre-displacements: One filled uncontrolled with a strong offset of $-3.96\,\textrm {m}^{-1}$ and two created with the controlled filling process, with $-$0.86 and $-0.75\,\textrm {m}^{-1}$, respectively. The lens with the lowest offset was able to deform also to positive focal powers, while the others were restricted to negative focal powers, at the advantage of an increased range of the spherical coefficient. Of the total number of finished lenses with the controlled filling process, half could be operated in both directions and half remained unidirectional. While there was no clear cutoff, the mean pre-displacements were -0.64 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ for the bidirectional lenses and -0.87 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ for the unidirectional lenses. These values are slightly greater than the overall mean in Fig. 1 as it includes an additional offset due to re-polarization, creep and hysteresis in operation.
Fig. 10. Outlines of the operating regions. Top left: Comparison of three typical lenses. Top right and bottom: The same three lenses in detail, with the pure bending and buckling mode shown for comparison. The symbols indicate the origin and the corner points of the voltage space.
Due to the strong offset, the bidirectional lens (top right) is stable at positive focal powers only with some contribution of the bending mode. In the pure buckling mode, it becomes unstable at some point and collapses to negative focal powers. For reference, we also show the pure buckling and bending modes. Ideally, the buckling should be identical to the corresponding segment of the overall trajectory and the bending curve should be enclosed by it. This is, however, not the case in practice due to creep and hysteresis. The focal power range is -6.8 to $5.6\,\textrm {m}^{-1}$ and the aspherical tuning range is of the order of 0.2 $\upmu$m over most of the focusing range.
The unidirectional lenses were stable in the full operating region, as expected. Again, there is clear evidence of hysteresis. The focusing power of the lens with small offset is equivalent to the negative branch of the bidirectional lens, while the spherical aberration can be tuned by 0.5 to 0.6 $\upmu$m over most of the focal powers. The strongly pre-deflected lens, however had a smaller focusing range from -2.8 to $-7.5\,\textrm {m}^{-1}$ due to the non-linearity of the buckling deformation and the range of the spherical aberration correction is less homogeneous. Hence, from a point of view of the operating region, there seems to be no particular advantage of this configuration.
While all fabricated lenses behaved slightly differently, they are all approximately represented by these three examples.
3.3 Dynamic behavior
To quantify the speed, we first measured the small signal (0.08 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ peak to peak) frequency spectrum of a lens with large offset and the lowest viscosity paraffin–PHF fluid, shown in Fig. 11, top left. In the low-frequency limit, we find a logarithmic decrease $a (1 - \gamma log_{10}(2\pi \, \nu \, \times \, 0.1\mathrm {s}))$ in the response with a coefficient of $\gamma \sim 0.08$, which is larger than typical values for piezoelectric creep. The first resonance lies at $4.0 \, \mathrm {kHz}$ with a q-factor (obtained from the amplitude) of 2.1 and the second resonance has a frequency of $8.2 \, \mathrm {kHz}$ with $q \sim 3.1$. Fitting the resonances to a damped resonator, we get eigenfrequencies of 4.3 and $8.3 \, \mathrm {kHz}$ and a decay time constant of $0.17\,\mathrm {ms}$ that is identical for both resonances, corresponding to quality factors of 2.3 and 4.4.
Fig. 11. Top left: Normalized small-signal frequency spectrum (lens with large offset and paraffin–PHF fluid). The horizontal grid lines are scaled proportional to a logarithmic fit in the low-frequency regime. Top right: Rising and falling large-signal step response for a step and approximated smooth exponential signals. The inset shows a fit to $a\, +\, b\, /\, \sqrt {t}$. Bottom left: Step response of a unidirectional lens with small offset and paraffin–PHPY fluid; combination of different step signals and bias, normalized to the largest response. The narrow lines are normalized to 1. Bottom right: Step responses of lenses filled with different fluids. The paraffin–PHF lens has a large offset and the others have a small offset. The narrow lines are normalized to the largest response.
This behavior agrees with the step response in Fig. 11 top right for which we kept one piezo sheet at 0 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ and applied a step of 1 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ to the other one. There, we find a strong resonance followed by a creep. The creep was slightly stronger in the falling edge than in the rising edge, but creep can typically be compensated by an appropriate active control. In the inset, we see that it actually follows a behavior proportional to $1/\sqrt {t}$, which is atypical and has yet to be understood. To illustrate how we can reduce the ringing, we drove the lenses also with an approximated (in three linear segments) exponential $1- 2^{-t/t_0}$, with half-times of 30 and $50\,\upmu \mathrm {s}$ and found a significant reduction in the ringing. The 90% response times were 76, 110 and $134\,\upmu \mathrm {s}$ for the rising edge and 65, 104 and $144\,\upmu \mathrm {s}$ for the falling edge, so we assume that we can achieve a 100 to $150\,\upmu \mathrm {s}$ response time without ringing using a more sophisticated driving signal.
To investigate the effect of bias and pre-displacement, we used a lens with small offset (the one shown in Fig. 10, bottom left) and applied all combinations of 1 ${{\frac {\mathrm {kV}}{\textrm {mm}}}}$ bias and steps to both actuators. In Fig. 11 bottom left, we see that in general, an actuation against the offset (blue and black curves) actually has a faster initial response – around $40\,\upmu \mathrm {s}$ – than the steps along the offset ($130\,\upmu \mathrm {s}$). This is probably a result of the combination of the wave propagating on the membrane and the dynamic pressure in the fluid. The step in the direction of the offset also shows a much larger (both absolute and relative) creep than the step against the offset. This indicates again that the creep is not primarily an effect of the piezo material, but rather results from the elastomer that is known to have a non-trivial dynamic response [9] and may be subject to non-linear geometric or material effects. It might also result from the viscosity of the fluid. Interestingly, the step response with bias showed less creep, probably because of the stronger force exerted by the pre-displaced sandwich membrane. We further compare the different fluids in Fig. 11, bottom right. We find that the fluid with the lowest viscosity shows the least creep and vice-versa.
Finally, we also performed a preliminary measurement of the long-term stability. Over 8 million cycles, we found no significant drift of the amplitude of the displacement. We found, however, a 1.7% day/night periodicity that probably results from thermal stress in the glass-piezo membrane. This temperature dependence and also the hysteresis and creep effects can be removed with appropriate sensors as it was shown in [26].
4. Summary and conclusions
In this paper, we have presented a varifocal lens with aspherical correction that we optimized for a large aperture to diameter ratio, achieving a ratio of 7.6 mm clear aperture to 9 mm unpackaged outer diameter. Using a fluid-membrane configuration with an active piezo-glass sandwich membrane first presented in [7], our key solutions were an optimized actuator geometry shown in section 2.1, a combination of optically matched fluids and transparent elastomer structures (sec. 2.2) with a compliant design and a novel two-step curing process in the fabrication that allowed for smaller contact pads (sec. 2.3). Yet, the different packages had 10 or 10.7 mm outer dimensions, and we did not use the outermost part of the clear membrane as some glue has flown onto the membrane during the dispensing. Both issues, however can be resolved in an industrial fabrication process with more accurate dispensing (or suitable hydrophobic coating of the clear segment of the membrane) and e.g. injection-molded packages.
Using an iterative controlled filling process, we found in section 3.1 an average RMS surface error of just 112 nm and offset of $0.07\,\textrm {m}^{-1}$ – before mounting the lens. After mounting and finishing, this offset increased to $0.57\,\textrm {m}^{-1}$, which can also be remedied in an industrial production with more precise elastomer lens chambers. A further improvement of the surface quality would require polishing or other treatments of the piezo material. A general improvement may also be cutting the piezo sheets with a water jet, diamond tools (thanks to a referee for pointing this out) or with an ultra short pulse laser.
The lenses operate in a combination of bending and buckling to control the value and sign of the focal power and the aspherical behavior (see sec. 3.2). Due to the large offset, not all lenses could deflect in both directions. The bidirectional lenses showed a focal power range of approximately $-$7 to $+6\,\textrm {m}^{-1}$ and the unidirectional lenses typically from $-$7 to $-0.5\,\textrm {m}^{-1}$, however at twice the range of the spherical Zernike coefficient. While we estimate that bidirectional lenses need a very low offset below $0.2\,\textrm {m}^{-1}$ to obtain a good aspherical control with a symmetric operating region, the unidirectional lenses may be controlled more easily and their state does also (ignoring hysteresis) not depend on the sequence of the operation.
The high stiffness of the membrane and the small moving mass provide a fast response. At the example of a lens with large offset, we found in section 3.3 a fist resonance at $4.0 \, \mathrm {kHz}$ and a 90% response time of $76\,\upmu \mathrm {s}$, however with significant ringing which could be reduced significantly with a smoother driving signal. Hence, we conclude that it is possible to achieve a usable response time around 100 to $150\,\upmu \mathrm {s}$ with an optimized filter for the driving signal or with active damping using the piezo actuators. As an alternative, we tried to reduce the ringing with more viscous lens fluids, which showed, however, no improvement. A future approach might be to integrate damping into the glass membrane by adding suitable coatings. The high-viscosity lenses also showed a relatively large creep, which has an untypical $1/\sqrt {t}$ dependence whose origin was not quite clear. A possibility is a viscoelastic behavior of the polyurethane elastomer, and approaches to address this are the use of alternative elastomers, thinner elastomer structures in an industrial fabrication or an active compensation with a suitable control.
Electronically, the lens acts as a capacitor with $\mathrm {nF}$ capacitance. Hence, it does not draw power when holding a constant focus, and the power consumption in dynamic operations can be reduced using driving electronics with charge recovery [27,28].
Overall, we conclude that we have shown the road towards the perfect varifocal lens with high-quality, active spherical aberration correction, exceptionally fast response time, low power consumption and minimized footprint. The aberration correction can not only help to improve the performance of optical systems but also reduce the complexity of the designs, enabling new highly compact and cost-efficient high-performance systems. The high speed can enable new scanning or tracking applications while the piezo actuation can minimize the power consumption. In [29], we had already demonstrated how even the un-optimized predecessor [15] of the present lens can provide the longitudinal scanning and aberration correction in confocal scanning microscopy. In the near future, we will present the use of the present lens in endoscopic microscopy, taking advantage of the compact size.
Funding
Deutsche Forschungsgemeinschaft (EXC1086, WA1657/6-1, WA1657/6-2); Deutsche Forschungsgemeinschaft and Albert-Ludwigs-Universität Freiburg (funding programm Open Access Publishing).
Acknowledgments
We would like to thank the Gisela and Erwin Sick chair for Micro Optics at IMTEK for access to their refractometer and Ziyu Wang for help with the refractometer measurements. We further thank Binal Bruno for help with the long-term measurement and Prof. Ulrike Wallrabe for comments on the manuscript.
Disclosures
The author declares no conflict of interest
References
1. H. Ren and S.-T. Wu, Introduction to Adaptive Lenses, vol. 75 (John Wiley & Sons, 2012).
2. H. Oku and M. Ishikawa, “High-speed liquid lens with 2 ms response and 80.3 nm root-mean-square wavefront error,” Appl. Phys. Lett. 94(22), 221108 (2009). [CrossRef]
3. OptotuneFast Electrically Tunable Lens (2019). Update: 09.08.2019.
4. J. Draheim, F. Schneider, T. Burger, R. Kamberger, and U. Wallrabe, “Single chamber adaptive membrane lens with integrated actuation,” in 2010 International Conference on Optical MEMS and Nanophotonics, (IEEE, 2010), pp. 15–16.
5. D.-Y. Zhang, V. Lien, Y. Berdichevsky, J. Choi, and Y.-H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82(19), 3171–3172 (2003). [CrossRef]
6. J. Jarosz, N. Molliex, G. Chenon, and B. Berge, “Adaptive eyeglasses for presbyopia correction: an original variable-focus technology,” Opt. Express 27(8), 10533–10552 (2019). [CrossRef]
7. M. C. Wapler, M. Stürmer, and U. Wallrabe, “A compact, large-aperture tunable lens with adaptive spherical correction,” in 2014 Int. Symp. Optomechatron. Technol. (ISOT), (IEEE, 2014), pp. 130–133.
8. L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensoen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Flexible lens assembly with variable focal length,” (2013). US Patent 8,390,939.
9. M. Wapler, C. Weirich, M. Stürmer, and U. Wallrabe, “Ultra-compact, large-aperture solid state adaptive lens with aspherical correction,” in 18th Int. Conf. Solid State Sens., Actuators and Microsyst. (TRANSDUCERS), (IEEE, 2015), pp. 399–402.
10. S. Kuiper and B. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004). [CrossRef]
11. S. Sato, “Liquid-crystal lens-cells with variable focal length,” Jpn. J. Appl. Phys. 18(9), 1679–1684 (1979). [CrossRef]
12. H.-C. Lin, M.-S. Chen, and Y.-H. Lin, “A review of electrically tunable focusing liquid crystal lenses,” Trans. Electr. Electron. Mater. 12(6), 234–240 (2011). [CrossRef]
13. K. A. Higginson, M. A. Costolo, and E. A. Rietman, “Adaptive geometric optics derived from nonlinear acoustic effects,” Appl. Phys. Lett. 84(6), 843–845 (2004). [CrossRef]
14. E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. (Melville, NY, U. S.) 102(3), 033104 (2007). [CrossRef]
15. F. Lemke, Y. Frey, B. P. Bruno, K. Philipp, N. Koukourakis, J. Czarske, U. Wallrabe, and M. C. Wapler, “Multiphysics simulation of the aspherical deformation of piezo-glass membrane lenses including hysteresis, fabrication and nonlinear effects,” Smart Mater. Struct. 28(5), 055024 (2019). [CrossRef]
16. M. C. Wapler and U. Wallrabe, “Ultra-fast and compact varifocal lens,” in 2019 IEEE 32nd International Conference on Micro Electro Mechanical Systems (MEMS), (IEEE, 2019), pp. 938–941.
17. M. C. Wapler, F. Lemke, G. Alia, and U. Wallrabe, “Aspherical high-speed varifocal mirror for miniature catadioptric objectives,” Opt. Express 26(5), 6090–6102 (2018). [CrossRef]
18. B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” Smart Mater. Struct. 28(1), 015029 (2019). [CrossRef]
19. H. J. Qi and M. C. Boyce, “Stress–strain behavior of thermoplastic polyurethanes,” Mech. Mater. 37(8), 817–839 (2005). [CrossRef]
20. A. Müller, M. C. Wapler, and U. Wallrabe, “A quick and accurate method to determine the poisson’s ratio and the coefficient of thermal expansion of pdms,” Soft Matter 15(4), 779–784 (2019). [CrossRef]
21. J. López-Gejo, J. T. Kunjappu, J. Zhou, B. Smith, P. Zimmerman, W. Conley, and N. J. Turro, “Polycycloalkanes as potential third-generation immersion fluids for photolithography at 193 nm,” Chem. Mater. 19(15), 3641–3647 (2007). [CrossRef]
22. W. F. Seyer and C. W. Mann, “The vapor pressures of cis-and trans-decahydronaphthalene,” J. Am. Chem. Soc. 67(2), 328–329 (1945). [CrossRef]
23. V. Belonenko, A. Gureev, and L. Lyshenko, “Bulk elastic properties of petroleum oils,” Chem. Technol. Fuels Oils 18(10), 510–514 (1982). [CrossRef]
24. SCHOTT North America, Inc., Product Information D 263 THIN BOROSILICATE GLASS (obtained 10/2019). Obtained at http://www.csimc-freqcontrol.com/data/upload/20180209/5a7d029ee790d.pdf.
25. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73(6), 860–861 (1983). [CrossRef]
26. H. G. B. Gowda and U. Wallrabe, “Integrated sensor system to control the temperature effects and the hysteresis on adaptive fluid-membrane piezoelectric lenses,” in 2019 Int. Symp. Optomechatron. Technol. (ISOT), (IEEE, 2019).
27. J. A. Main, D. V. Newton, L. Massengill, and E. Garcia, “Efficient power amplifiers for piezoelectric applications,” Smart Mater. Struct. 5(6), 766 (1996). [CrossRef]
28. D. Campolo, M. Sitti, and R. S. Fearing, “Efficient charge recovery method for driving piezoelectric actuators with quasi-square waves,” IEEE Trans. Sonics Ultrason. 50(3), 237–244 (2003). [CrossRef]
29. K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue with an aberration-correcting adaptive lens,” Sci. Rep. 9(1), 9532 (2019). [CrossRef]
