Abstract

We present a varifocal mirror based on a piezo-actuated glass membrane that can be used as a secondary mirror in miniature Cassegrain-type mirror- or catadioptric objectives. The mirror section has a diameter of 10 mm on a clear membrane diameter of 23 mm, with a focal range of ±8 m−1 and a response time on the millisecond-scale. The two piezo layers enable an aspherical tuning range that covers the elliptic, parabolic and hyperbolic regime over most of the focal range. We demonstrate the application of the mirror in a simple catadioptric telefocus objective with a focal length of 68 mm at an aperture of 22 mm and a thickness of 16.6 mm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Mirror objectives are, on the one hand, much lighter than refractive objectives and, on the other hand, the folded lightpath reduces the length of the optical system. Purely reflective objectives also avoid chromatic aberrations. The two most significant drawbacks are the inability or difficulty to adjust the aperture and a high susceptibility to stray light. Furthermore, they have annular, rather than circular, defocus artifacts (“bokeh”), which may be considered less aesthetic in artistic images. Hence, they are most commonly used in large telescopes, space-based optics and some large telefocus camera objectives. They are furthermore used in long working distance microscope objectives.

Another possible application are miniature telefocus objectives: While modern miniature objectives can reach large numerical apertures to image on high resolution sensors, they are usually limited to short focal lengths, i.e. small geometric apertures. Typical mobile cameras with 7 mm distance from the first lens apex to the image plane have 4 to 5 mm focal length and ≈2 mm aperture [1]. Hence, they provide wide angle images and have a large depth of field. Reflective or catadioptric objectives can circumvent this problem and enable larger focal lengths and larger apertures by folding the lightpath, in extreme cases 4 to 8 times [2,3]. Yet, there is one problem with the miniaturization of reflective objectives: The typical method of focusing miniature optics is shifting the whole lens barrel by electromagnetic [4] or – less frequently – piezo [5] actuation. On the one hand, the wider objectives become heavier, reducing their focusing speed and making them more sensitive to vibrations. On the other hand, the shift required to obtain a certain near focal point increases proportional to the square of the focal length, Δzshift ~ f2/znear. The other focusing mechanism, adaptive lenses, e.g. [6,7], becomes problematic at large apertures as they are typically several times larger than their usable aperture.

In this paper, we present a novel approach of focusing miniature reflective optics. Rather than shifting optical elements, we deform the secondary mirror. Hence, we require a large focal power in the regime of 10 m−1, an aperture of several mm and ideally no optical obstruction outside of the aperture of the mirror.

Most current deformable mirrors aim at higher order aberration correction in high-precision optical systems such as astronomical telescopes [8] and microscopes [9]. In addition to reflective liquid crystal spatial light modulators, there are two commercially available types of adaptive mirrors: Electrostatically driven membrane mirrors use a set of MEMS electrostatic actuators that deforms a thin membrane, see e.g. [10,11]. Piezoelectric deformable mirrors are either thick membranes that are deformed by the longitudinal deformation of a set of stack actuators [12] or they have, more commonly, a piezoelectric disk with a large number of electrodes bonded to a mirror-coated stiff substrate such as glass or silicon, using the bending effect of the transverse contraction of the piezo material and the passive substrate [13,14]. Both have in common a bulky construction and a small focal range, e.g. 2 m−1 [14]. The mirror that comes closest to our system is a micro mirror with 300 µm aperture that is based on a a MEMS-fabricated 6 µm thick Si/SiO2 mirror membrane that is deformed with a 3 µm thick annular in-plane polarized piezo actuator and is illuminated from the backside through a hole in the silicon substrate [15]. Yet, this is far away from our requirements.

In contrast, we have deposited a mirror in the inner part of a thin glass-piezo composite membrane as we will describe in section 2.1. This mirror segment then acts as a varifocal secondary mirror, while the piezo materials are located outside of the lightpath of the incoming light. We counter-act possible distortions of the membrane both in our fabrication process and by a stiff frame with a flexible mounting system described in section 2.2. The two degrees of freedom allow for a simultaneous control of the focal power and the spherical aberration as we will investigate in section 3, and we will further see that the axial shift of the mirror enhances the focusing effect. While this approach may be slower than deforming only the mirror section, it is still faster than other focusing techniques. Our mirror section has a diameter of 10 mm with a clear membrane diameter of 23 mm. We demonstrate its application in a simple catadioptric objective with 68 mm focal length then focused to infinity and 22 mm aperture at an overall axial length of 16.6 mm that uses a radially segmented lens and that we describe and characterize in section 4.

2. Design and fabrication

We use the design of the active glass-piezo membrane first described in [16] and [17] in the context of varifocal lenses.

2.1. Membrane design and simulation

The active membrane consists of 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. 1:

  • The mean strain of both (assumed to be narrow) piezo sheets with radius R,
    ΔR=Rd3112(Elow+Eup)
    controls the Dirichlet boundary condition of the membrane. Keeping in mind that the transverse piezo coefficient d31 is negative, mean electric fields of the sheets in the direction of the polarization, Elow + Eup > 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 where we assume a spherical deformation and neglect the mechanical response of the glass membrane, the curvature of the membrane becomes
    ρcurv.1~R1d313(Elow+Eup).
  • The difference of the strains, in contrast, will cause a bending deformation. Ignoring forces such as the mechanical backreaction 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,
    rzmembrane~Rhd31(ElowEup).

 

Fig. 1 Schematic drawing of the buckling (left) and bending deformation (right).

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In an assumed spherical deformation, this gives a curvature radius of

ρcurv.1~h1d31(ElowEup).
In combination, the boundary conditions then allow us to control both the curvature of the membrane at its center and the next subleading (quartic) term in the deflection profile, i.e. the focal power and the spherical aberration.

As these estimates to not take into account forces at all, we also performed an FEM simulation of the membrane deflection in different actuation modes using COMSOL. The aim is not a full optimization, but a parametric overview in order to find the best width of the piezo ring for a selection of possible material substrates and to determine the influence of the elastic suspension. To obtain a highly resolved and fast result, we use a rotationally symmetric model with fine meshing as shown in Fig. 2, and we enabled the non-linear deformations. In the multistable pure buckling mode, COMSOL still does not find one of the two global minima when starting from an undeflected initial state. Hence, we simulated sequentially first the bending mode with (0.3,0.3)kVmm, a mixed mode with (1.2,0.3)kVmm, and then two pure buckling modes with (1.2,1.2)kVmm and (0.3,0.3)kVmm, taking always the previous result as a starting point. For the glass membrane, we used the manufacturer data of SCHOTT D263t [18] with a Young’s modulus of 72.9 GPa, a Poisson ratio of 0.209 and three thicknesses hglass, 50 µm, 70 µm and 100 µm. Since the piezo coefficients tend to be highly non-constant [19], we simply used the built-in PZT-5H with d31 = −274 pm/V and a Young’s modulus of 127 GPa. Restricted by our available raw piezo sheets, we simulated the thicknesses hPZT = 100 µm and 150 µm, an outer radius Rout of 13 mm and inner radii Rin of 10 and 11 mm. Finally, we used the polyurethane Smooth-On ClearFlex 50 for the elastomer suspension ring, for which we measured a Young’s modulus of approximately 2.5 MPa and assumed a Poisson ratio of 0.45. The ring is placed at the inner diameter of the PZT ring as shown in Fig. 2 and keeping the thickness constant at 1 mm, we simulated different widths wsusp to investigate its influence. To analyze the results, we fitted the polynomial

zα0+12(,12f1r2+α4r4+α6r6++α12r12),
where f gives the focal length and α4 gives the aspherical Zernike coefficient of the reflected wave front, upon suitable normalization with the aperture A and wavelength λ:a4 ,0=A465λα4. At this point, it is suitable to expand to such high orders because we want to make sure that the first two orders are not affected by the profile in the outer region. As a further criterion, we define the usable aperture radius R0 at which the local second derivative of the profile deviates by 20% from the value at r = 0.

 

Fig. 2 Left: Drawing and simulation result of the mirror membrane at (1.2,1.2)kVmm, displacement scaled by a factor of XXX. Right: Closeup of the meshing at the contact of the glass membrane (blue shading), piezo rings (red) and polyurethane suspension (gray).

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We see in table 1 first of all that the width of the suspension ring has little influence on the displacement profile. There is a slight reduction in the bending mode at the widest ring, and a small increase in the buckling mode. The latter is consistent with a resistance against bending, as a flat profile at the side enhances the buckling effect at the center of the membrane. The thicker piezo actuators decrease the focal power in the bending mode as expected by eq. (4). They also increase the focal power in the buckling mode, on the one hand, because they can simply exert a larger force on the glass membrane and, on the other hand, because they also resist the deflection at the sides. Looking at the increasing glass membrane thickness, we see overall a change from elliptic profiles with α4 > 0 towards more hyperbolic profiles with α4 < 0. This may be due to the higher stiffness that causes a more even curvature distribution in the bending mode and a shift of the inflection point of the profile towards smaller radii in the buckling mode as shown in Fig. 3. The high stiffness of the 100 µm thick membranes also causes the significant drop in displacement in the buckling mode that has a highly non-linear behavior of the forces at small displacements, causing a threshold for the buckling to occur as we see in the buckling mode at 30 V. The aspherical coefficient also shows that we will always expect the profile in the buckling mode to become hyperbolic with decreasing fields. Finally, the larger focal power of the smaller clear aperture in particular in the buckling mode may result from the wider and hence stronger piezo rings and also from the geometry by eq. (2). For our prototype, we choose the 70 µm thick glass membrane with the 150 µm thick piezo film and the 22 mm clear aperture, as this provides a strong focal power in the buckling mode, large R0 and an only slightly elliptic profile at maximum buckling that will provide a tuning range into hyperbolic profiles at lower buckling. The 22 mm diameter will also allow us a larger aperture of the demonstrator objective than the 20 mm inner diameter, which would have given a larger focal power.

Tables Icon

Table 1. Summary of the simulation: The focal power and aspherical coefficient of different membrane designs in the bending mode (0.3,−0.3) kV/mm, combined bending and buckling (1.2,−0.3) kV/mm and buckling mode at low fields (0.3,0.3) kV/mm and high fields, (1.2,1.2) kV/mm. The usable aperture R0 is only shown for the pure bending and high-field buckling modes. The numbers are colored on a rainbow scale from red (lowest) to blue (highest).

 

Fig. 3 Left: Displacement of the mirror membrane for Rin = 11mm, hPZT = 150µm and different glass membrane thicknesses in the bending ((0.3,0.3)kVmm) and buckling ((1.2,1.2)kVmm) mode. Right: Local curvature (second radial derivative). The inset shows the buckling mode at large radii and the circles indicate R0; the 100 µm membrane had R0 > 0 in the bending mode.

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2.2. Overall design and fabrication

In the final design, we did some small modifications. For manufacturing reasons, we chose a suspension ring of 1.4 mm height and 0.6 mm width. We further increased the inner diameter of the piezo ring to 23 mm and the outer diameter to 27.5 mm in oder to maximize the aperture, keeping a margin of error of 0.5 mm e.g. for displaced glue at each side, and to maximally use the available piezo materials.

In order to contact and integrate the mirror in an optical system, the membrane is mounted on a 0.5 mm thick PCB as shown in Fig. 4. On the one hand, this carries contact pads for the external electric connections and, on the other hand, it has also four mounting points that are connected with spring segments without copper layer to relax possible mechanical distortions. For this reason, we also stiffen the element with a 0.7 mm thick silicon ring.

 

Fig. 4 Schematic drawing of the mirror membrane (left) and the full mirror system (right).

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The fabrication process is summarized in Fig. 5. We first laser-structure the PZT sheets and glass membrane, on which we deposit a chromium/aluminum mirror layer using a shadow mask. Then, we glue the glass-piezo sandwich membrane in two steps using Resoltech HTG 240 high-Tg epoxy glue and annular vacuum chucks made from polished silicon wafers that carry the glass membrane. The aim is to avoid distortions in the membrane from the uneven piezo sheets and to avoid damage to the mirror-coated region. To prevent a pre-deflection from thermal strains, we keep the curing temperature low at 35°C. In principle, one could compensate the thermal stress during the curing by applying an appropriate voltage to the piezo films. We contact the piezo sheets on their contact pads using low-temperature solder and insulate the outer edges of the piezo sheets using ClearFlex 50 polyurethane to avoid electrostatic breakdown.

 

Fig. 5 Illustration of some fabrication steps: Glueing of the membrane in two steps using vacuum chucks (a), fabrication of the PDMS mold (red) of the elastomer ring using a positive master (b), casting the polyurethane elastomer ring (c) and final assembly (d).

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To provide a flat carrier structure, we first hot-emboss the laser-structured FR4 PCB in a precision press at approximately 150°C and add the stiffening ring that is laser-structured from a silicon wafer. The polyurethane suspension ring is then molded directly onto the PCB using a PDMS mold that is created with a positive master made from silicon wafers and FR2. Finally, we glue the sandwich membrane onto the support structure using the same polyurethane and again the vacuum chuck to hold the membrane.

3. Characterization

Since all of the optical information of the mirror is contained in the surface profile, we only measure its mechanical deformation and derive the focal length and aspherical coefficient. We use and optical profilometer that scans the membrane pointwise on a 10 × 10 mm2 surface with a 250 µm grid during a periodic actuation with one period per point. To avoid edge effects on the mirror-coated region, we only take into account the central 9 mm diameter. We correct for any misalignment and tilt by fitting a 4th order polynomial h(x, y) = a0 + ax x + ay y + a2(x2 + y2) + a4(x2 + y2)2 to the surface with an estimated central point and then use the data of a whole actuation cycle to obtain the constant shift and tilt that best fits the data.

To visualize the aspherical behavior in two different ways, we used one “maximal” voltage trajectory that approximately outlines the stable operating region and another one that systematically scans it. In the maximal trajectory, we start with a state in which the membrane is put under tension, (0.3,0.3)kVmm, then bend it at its boundary as we go to (0.3,1)kVmm, compress it into a buckling mode at (1,1)kVmm, relax it back to (0.3,0.3)kVmm and repeat the cycle in the opposite direction. In principle, there is also a metastable operating region, where we start in the buckled state, e.g. (1,1)kVmm, and then bend the edge in the opposite direction, in our example (1,1x)kVmm, until we reach the point of instability where the membrane flips towards the other side.

Looking first at the pure bending and buckling modes in Fig. 6, we see the approximately linear behavior in the bending mode and the highly non-linear behavior in the buckling mode as already observed in [16, 17]. In the bending mode, we see a small pre-deflection, and we see that the simulation with the standard d31=274pmV approximately describes the behavior. In the buckling mode, we see a significant deviation between the measurements carried out in a periodic buckling-only actuation between 0.3 and 1kVmm in the upwards and downwards directions and the corresponding segments of the maximal trajectory. This illustrates the strong effect of the history of the piezo actuator. All curves show a behavior similar to a square root law as in eq. (2) and in addition a threshold electric field for buckling to occur which is significantly higher in the simulationd than in the experiments even if we increase the charge coefficient of the piezo to a realistic value of 400pmV [19]. This may arise from the assumption of ideal symmetry in the simulation, whereas the prototype has a pre-deflection, surface distortions and local defects that may trigger buckling already at smaller applied fields.

 

Fig. 6 Focal power as a function of the applied electric field at the upper actuator in the bending mode (left, Elow = −Eup) and buckling mode (right, Elow = Eup). The measurements in the buckling mode include the positive and negative branch in a periodic buckling-only actuation and the decreasing electric field in the maximal trajectory.

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In the maximal trajectory in Fig. 7, we see that the mirror can cover both the hyperbolic and elliptic surface profiles, and the ideal parabola over most of the focal range. The simulation using a charge coefficient of 400pmV resembles the measurement very closely. The fact that the regions of high voltages of the maximal trajectory are reproduced best with the higher charge coefficient and the bending mode at low fields is best described with a smaller charge coefficient agrees with the typical non-linear behavior of PZT ceramics that have an increasing charge coefficient with increasing applied fields, with a maximum near the coercive field strength [19]. The slightly higher overall focal power in the measurement may be a result of small misalignments and glue flowing onto the membrane at the inner edge of the piezo rings. This reduces the effective diameter of the glass membrane and hence increases the focal power according to table 1. We verified this by repeating the simulation with an inner diameter of 22 mm, which shows a larger focal power, so some fabrication uncertainty of 0.2 to 0.5 mm can most likely account for the deviation.

 

Fig. 7 Quartic coefficient in the reflected wavefront as a funtion of the focal power for the maximal trajectory described in the text. The top and right axes indicate the F-number (assuming an aperture of 9 mm) and the aspherical Zernike coefficient normalized to the d-line.

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We further show the full dependence of the mirror profile on the applied electric fields in Fig. 8. In order to stay in the stable regime with a defined buckling direction, we increased one voltage in steps of 10 V (0.067kVmm) and swept the other voltage for each step starting from this value up to 1kVmm. This covers all combinations in two regimes of Eup > Elow and Eup < Elow. We see again a small pre-deflection and asymmetry. We further see that the focal power is dominated by the buckling deformation as the contours are approximately diagonal, i.e., parallel to Eup + Elow = const.. The aspherical deformation is predominantly controlled by the bending effect. On the right, we show the RMS wavefront error, which we define as the deviation from rotational symmetry. The smallest error is just above 500 nm and it increases in the buckling regime where both actuators contract, whereas a strong bending effect or expanding piezo rings seem to stabilize the membrane. The linear shift of the wavefront, i.e. twice the displacement of the membrane, ranges approximately from −400 to 400 µm.

 

Fig. 8 Left: Focal power (contour lines) and aspherical coefficient (colors) as a function of the applied voltage. Right: Linear shift of the reflected wavefront at its center relative to (0.3,0.3)kVmm (contour lines) and RMS wavefront error (colors).

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Let us finally look at the resonance spectrum and the step response in Fig. 9 which we measured both at a single point in the center using a laser triangulation sensor. We obtained the resonance spectrum with a driving signal of ±0.003kVmm in the pure bending mode, without a pre-deflection and with a buckling pre-deflection at a bias field of 0.67kVmm. The first resonance occurs at 1.32 kHz, which increases to 1.7 kHz with the pre-deflection, i.e. the deflected membrane becomes stiffer. In a critically damped system, this would allow for a sub-ms step response. Our system, however, is highly underdamped with a quality factor of 14 without deflection and 19 with pre-deflection. This can be seen in the step response on the right, where we applied a 0.67kVmm field to one piezo sheet, i.e. in a combined bending/buckling displacement. There, we see a strong ringing, corresponding to the high Q factor. To illustrate the potential to suppress the ringing, we also applied linearly rising and falling signals of different rise/drop times, which reduced the ringing to below 10%. We expect that a more sophisticated driving signal or filter could suppress the ringing even further. The creep of the piezo material is apparent in the step response, where the 0.5 s levels are approached from below in the rise and from above in the drop, and also in the resonance spectrum that shows a small decrease in the response up to 200 Hz. The smaller response of the pre-deflected membrane in the frequency sweep simply results from the non-linear displacement of the buckled membrane.

 

Fig. 9 Left: Frequency spectum (RMS displacement) of a ±0.033kVmm bending signal without bias and 0.67kVmm buckling bias, normalized to the response without bias at 10 Hz. Right: Response to a rising (solid) and falling (dashed) 0.67kVmm step applied to one piezo (mixed bending/buckling), normalized to the displacement at 0.5 s. The times indicate the rise/drop time of the linear rising and falling edge.

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4. Imaging demonstrator

To illustrate the operation of the mirror in an imaging application, we built a simple doubly-folded catadioptric objective, consisting of two lenses and a flat primary mirror. We chose a 22 mm aperture and a 30 mm × 30 mm board-level camera with a 1/2.5” Aptina MI5100 color sensor with 2.2 µm pixels.

4.1. Fabrication and optical design

In principle, a doubly-folded optical design has discrete regions for the incoming and reflected light near the mirrors and overlapping light paths in the center. For an economical design that produces a minimum of imaging quality, we chose a polymer asphere made from E48r with diffractive color correction as a primary lens. To demonstrate the optical design constraints and and create one more degree of freedom, we hot-embossed a different curvature at the center of this lens as shown in Fig. 10(a). On one (upper) side, we use a plano-spherical glass lens as a mold that is horizontally stabilized and on the other (lower) side, we embed the polymer lens in PDMS. During the embossing, we keep the lower surface at room temperature and heat the upper surface and hence the molding lens to 170°C. We deposit the primary mirror (Cr/Al) on the backside of a second glass lens (eventually N-BK7) using a shadow mask. To prevent ghost images and reduce stray light, we further place a laser-structured annular aperture of 9/11 mm around the embossed surface that can be seen at the center of the lens in Fig. 10(b), in addition to the primary 22 mm aperture.

 

Fig. 10 (a) From left to right: Polymer lens in the embossing mount with PDMS support (red), with glass lens as negative mold and finished lens. (b) Objective structure with fixed lens and primary mirror, complete system without primary aperture.

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We optimized the optical design in a sequential ray tracing model using Zemax, putting appropriate constraints to represent one physical surface by up to 3 surfaces in the model. To create a telefocus system, we initially constrained the focal length and overall length of the system and relaxed them only at the end in the final optimization step. In the meantime we tried different available aspheres and first kept the surfaces of the glass lens variable, replacing the surfaces later with values of different stock lenses. The final design as shown in Fig. 11 was optimized to L0 = 1700 mm object distance with a non-actuated secondary mirror such that we can focus to ∞ with ≈6 m−1 at the secondary mirror. The focal length of the objective was 57 mm, or 68 mm when focused to infinity, which is approximately 4 times the 16.6 mm axial length of the optical system from the aperture/secondary mirror to the image plane. In the spot diagram on the right, we see that the resolution with an RMS spot radius ranging from 25 to 40 µm is far from optimal, but it will be sufficient to demonstrate some imaging and focusing.

 

Fig. 11 Left: Ray tracing diagram of the imaging system. Right: Spot diagrams with three different wavelengths in the center, at 1 mm radius (upper edge of the image) and 2 mm radius (far corner of the sensor). The grid lines are in steps of 20 µm.

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4.2. Characterization

Due to the surface coating of the lenses, it was not possible to measure the actual profile of the polymer lens after embossing. In first tests, the focus with a flat mirror was at approximately 930 mm, which is equivalent e.g., to a change of the radius of curvature of the embossed section from −9.3 mm to −10.2 mm. As this moved infinity outside the focal range, we shifted the polymer lens by 350 µm towards the glass lens, which shifted the focus to approximately L0 = 1400 mm and hence put infinity back into the focal range.

The focusing consists both of a change in focal power of the mirror and a shift of the wavefront Δlopt due to a shift of the mirror. It turns both effects are approximately proportional, such that the mirror shift can be represented by an additional focal power

Δf1=ϕΔlopt,ϕ14.00×103m2
near f−1 = 0 and with a non-linearity of up to 13% at Δlopt = 200 µm. This allows us to define an effective focal power feff1 that includes the mirror shift and has in our case a range of ±13.6 m−1.

To measure the resolution and the applied voltages for the focusing on different planes at distances L, we produced line patterns as shown in Fig. 13(a), which we imaged at different distances with a set of voltages near the subjective optimal focus and with a chart size that appeared to range from reasonably sharp to almost completely unsharp. To obtain an approximately parabolic profile, we chose voltage combinations near the combined bending/buckling branch of Fig. 7, i.e. near the outer perimeter of Fig. 8. We then segmented the images and obtained the contrast ratio from the amplitude of the peaks in the Fourier spectra taken in the direction of the pattern. To find the resolution, we fitted a gaussian decay aek2/σ2 to the contrast ratio and defined the resolution as the line pairs per mm of the 1/e drop, σ, in the contrast. We defined the optimal focus as the voltage combination that gave the highest resolution and reconstructed the effective focal power from an interpolated fit to Fig. 8.

 

Fig. 12 Left: Focal power of the mirror for different image planes and the thin lens fit described in the text. The measurement data includes both the effective focal power feff (curvature and shift) and the contribution of the curvature of the mirror. Right: Measured and simulated resolution of the imaging system. The line indicates the resolution at 1400 mm scaled by the distance assuming constant magnification.

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Fig. 13 Left: Resolution test chart and image taken at 3 m distance (cropped to 52% and scaled in brightness) Right: Images of a watch at 0.59 m distance and a roof with a clock tower at 110m distance obtained at the two different foci. Both objects are present in both images.

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While in principle, this system with effectively 8 surfaces is non-trivial to describe, the focusing effect of the mirror can be well approximated by an ideal lens combination,

L1~ρ1feff1+L01.
The factor ρ ∼ 19.4 describes the reduction of the focusing effect of the varifocal mirror in this optical system compared to e.g., a varifocal lens placed next to an ideal thin lens. In left of Fig. 12, we see that this is a good approximation over the whole focal range. Comparing the data points of f and feff, we also see that the mirror shift contributes more than 1/3 to the effective focal power. The measured focal power also agrees well with the simulation results, roughly within the estimated uncertainties that are relatively large due to the limited optical quality of the objective. There is only a significant deviation in the measurement at 600 mm distance. We assume that this may be due to the decreasing magnification with the decreasing focal length of the system at short object distances, which may make the chart that is slightly out-of-focus appear in focus because of a higher magnification. In the right graph, we see that the observed resolution is approximately 20% lower than the simulation result that we obtained from the modular transfer function. This is not likely to be related to the varifocal mirror, as its quality is, on the one hand, in the range of 1λ at low foci and we would, on the other hand, also expect to see a worsening quality also for the 10,000 mm image. Instead, it is more likely to result from deformations of the aspherical lens during the embossing or simply from some measurement artifact. Again, the measurement at 600 mm shows a significant deviation.

As an illustration, we finally show an image of the line chart and images with different foci of objects at 0.59 m (watch) and 110 m (clock tower) distance in Fig. 13(b). We obtained the images of the charts with spotlight illumination in a dark room and used an approx. 20 cm long lens shade to reduce stray light in the clock/watch images. While there are stray light and shading artifacts in the corners, we clearly observe the short depth of focus due to the large aperture and the large zoom factor compared to ordinary miniature cameras.

5. Summary, discussion and conclusions

In this paper, we showed a novel varifocal mirror with active aspherical correction based on a piezo-glass-piezo composite sandwich membrane that is particularly suitable for focus control in Cassegrain-type mirror- or catadioptric objectives. In section 2, we developed a membrane configuration with a clear aperture of 23 mm and a mirror section of 10 mm diameter, at just 27.5 mm outer diameter of the annular piezo actuators for which we expect a wide aspherical tuning and focusing range. In principle, the diameter of the mirror section could be changed to any diameter within the clear aperture, making it suitable also for other applications. The overall design and fabrication aims to minimize the distortion of the mirror membrane.

We saw in section 3 a wavefront error between 0.5 and 1.2 µm, a focusing range of more than ±8 m−1 and an asperical control that covers positive and negative aspherical coefficients over most of the focusing range. The fact that the range of the aspherical parameter vanishes approximately linearly with decreasing focal prower is not critical in typical applications as the fixed part of the optical system can be optimized for this point. The resonance frequency of more than 1.3 kHz shows the potential for sub-ms response times. Achieving these, however, will require additional damping, a sophisticated driving signal or a suitable electrical filter to suppress the ringing of the membrane.

In section 4, we demonstrated a possible imaging system based on just two lenses. To show potential design features, we hot-embossed a different curvature at the center of one lens and deposited the primary mirror on the second lens. Overall, we achieved a maximum focal length of 68 mm and aperture of 22 mm at just 16.6 mm length of the optical system. The optical quality and focusing response was reasonably close to the predictions of the simulation. Still, this system is only a demonstrator that is far from diffraction limited.

To conclude, we have shown how to fabricate compact, high-speed varifocal mirrors with active aspherical correction and large apertures. These enable focusing in catadioptric objectives but can also be used for other applications. A miniaturization of the mirror in the presented design is possible by a factor of at least 3, further miniaturization may require a change of the fabrication to MEMS techniques. A challenge will be the development of the corresponding miniaturized high-quality catadioptric objectives with low stray light that have the potential to enable miniature telefocus imaging systems with large apertures. Even only our demonstrator scaled by 1/2.4 to the 7 mm length of a mobile camera would give a 7× zoom factor and a 4× larger aperture than ordinary lens systems.

Funding

German research foundation (DFG) grant number WA1657; Cluster of Excellence “BainLinks-BrainTools” (DFG grant EXC1086).

Acknowledgments

We would like to thank Binal Bruno for his help in fabricating a prototype for an earlier approach that was not realized eventually.

References and links

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2. E. J. Tremblay, R. A. Stack, R. L. Morrison, and J. E. Ford, “Ultrathin cameras using annular folded optics,” Appl. Opt. 46, 463–471 (2007). [CrossRef]   [PubMed]  

3. E. J. Tremblay, R. A. Stack, R. L. Morrison, J. H. Karp, and J. E. Ford, “Ultrathin four-reflection imager,” Appl. Opt. 48, 343–354 (2009). [CrossRef]   [PubMed]  

4. C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009). [CrossRef]  

5. H.-P. Ko, H. Jeong, and B. Koc, “Piezoelectric actuator for mobile auto focus camera applications,” J. Electroceram. 23, 530 (2009). [CrossRef]  

6. A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012). [CrossRef]  

7. L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct. 25, 2011). US Patent 8,045,280.

8. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999). [CrossRef]  

9. M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365, 2829–2843 (2007). [CrossRef]  

10. T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999). [CrossRef]  

11. E. J. Fernández and P. Artal, “Membrane deformable mirror for adaptive optics: performance limits in visual optics,” Opt. Express 11, 1056–1069 (2003). [CrossRef]   [PubMed]  

12. R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008). [CrossRef]  

13. I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007). [CrossRef]  

14. Oko guide to adaptive optics,” OKO Technologies (2013).

15. M. J. Mescher, M. L. Vladimer, and J. J. Bernstein, “A novel high-speed piezoelectric deformable varifocal mirror for optical applications,” in “15th IEEE Int. Conf. Micro Electro Mech. Syst. (MEMS),” (IEEE, 2002), pp. 511–515.

16. 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.

17. 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.

18. D263t eco specification - physical and chemical properties,” SCHOTT Advanced Materials (2013).

19. B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

References

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  1. T. Steinich and V. Blahnik, “Optical design of camera optics for mobile phones,” Adv. Opt. Technol. 1, 51–58 (2012).
  2. E. J. Tremblay, R. A. Stack, R. L. Morrison, and J. E. Ford, “Ultrathin cameras using annular folded optics,” Appl. Opt. 46, 463–471 (2007).
    [Crossref] [PubMed]
  3. E. J. Tremblay, R. A. Stack, R. L. Morrison, J. H. Karp, and J. E. Ford, “Ultrathin four-reflection imager,” Appl. Opt. 48, 343–354 (2009).
    [Crossref] [PubMed]
  4. C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
    [Crossref]
  5. H.-P. Ko, H. Jeong, and B. Koc, “Piezoelectric actuator for mobile auto focus camera applications,” J. Electroceram. 23, 530 (2009).
    [Crossref]
  6. A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
    [Crossref]
  7. L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.
  8. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
    [Crossref]
  9. M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365, 2829–2843 (2007).
    [Crossref]
  10. T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
    [Crossref]
  11. E. J. Fernández and P. Artal, “Membrane deformable mirror for adaptive optics: performance limits in visual optics,” Opt. Express 11, 1056–1069 (2003).
    [Crossref] [PubMed]
  12. R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
    [Crossref]
  13. I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007).
    [Crossref]
  14. “Oko guide to adaptive optics,” OKO Technologies (2013).
  15. M. J. Mescher, M. L. Vladimer, and J. J. Bernstein, “A novel high-speed piezoelectric deformable varifocal mirror for optical applications,” in “15th IEEE Int. Conf. Micro Electro Mech. Syst. (MEMS),” (IEEE, 2002), pp. 511–515.
  16. 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.
  17. 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.
  18. “D263t eco specification - physical and chemical properties,” SCHOTT Advanced Materials (2013).
  19. B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

2012 (2)

T. Steinich and V. Blahnik, “Optical design of camera optics for mobile phones,” Adv. Opt. Technol. 1, 51–58 (2012).

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

2009 (3)

E. J. Tremblay, R. A. Stack, R. L. Morrison, J. H. Karp, and J. E. Ford, “Ultrathin four-reflection imager,” Appl. Opt. 48, 343–354 (2009).
[Crossref] [PubMed]

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

H.-P. Ko, H. Jeong, and B. Koc, “Piezoelectric actuator for mobile auto focus camera applications,” J. Electroceram. 23, 530 (2009).
[Crossref]

2008 (1)

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

2007 (3)

I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007).
[Crossref]

M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365, 2829–2843 (2007).
[Crossref]

E. J. Tremblay, R. A. Stack, R. L. Morrison, and J. E. Ford, “Ultrathin cameras using annular folded optics,” Appl. Opt. 46, 463–471 (2007).
[Crossref] [PubMed]

2003 (1)

1999 (1)

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[Crossref]

Artal, P.

Bernstein, J. J.

M. J. Mescher, M. L. Vladimer, and J. J. Bernstein, “A novel high-speed piezoelectric deformable varifocal mirror for optical applications,” in “15th IEEE Int. Conf. Micro Electro Mech. Syst. (MEMS),” (IEEE, 2002), pp. 511–515.

Bifano, T. G.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[Crossref]

Bihan, D.

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

Blahnik, V.

T. Steinich and V. Blahnik, “Optical design of camera optics for mobile phones,” Adv. Opt. Technol. 1, 51–58 (2012).

Bolis, S.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Booij, W.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Booth, M. J.

M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365, 2829–2843 (2007).
[Crossref]

Bouvier, C.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Bridoux, C.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Bruno, B. P.

B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

Chang, Y.-H.

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

Dournaux, J.

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

Eliassen, M.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Fahmy, A. R.

B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

Fanget, S.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Fernández, E. J.

Ford, J. E.

Gasmi, R.

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

Hammer, F.

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

Haugholt, K. H.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Henriksen, L.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Horenstein, M. N.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[Crossref]

Horng, J.-B.

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

Jacquet, F.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Jagourel, P.

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

Jeong, H.

H.-P. Ko, H. Jeong, and B. Koc, “Piezoelectric actuator for mobile auto focus camera applications,” J. Electroceram. 23, 530 (2009).
[Crossref]

Johansen, I.-R.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Kanno, I.

I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007).
[Crossref]

Karp, J. H.

Kartashov, V.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Ke, S.-S.

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

Ko, H.-P.

H.-P. Ko, H. Jeong, and B. Koc, “Piezoelectric actuator for mobile auto focus camera applications,” J. Electroceram. 23, 530 (2009).
[Crossref]

Koc, B.

H.-P. Ko, H. Jeong, and B. Koc, “Piezoelectric actuator for mobile auto focus camera applications,” J. Electroceram. 23, 530 (2009).
[Crossref]

Kopp, C.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Kotera, H.

I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007).
[Crossref]

Kunisawa, T.

I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007).
[Crossref]

Lin, P.-D.

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

Lin, P.-H.

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

Liu, C.-S.

C.-S. Liu, P.-D. Lin, P.-H. Lin, S.-S. Ke, Y.-H. Chang, and J.-B. Horng, “Design and characterization of miniature auto-focusing voice coil motor actuator for cell phone camera applications,” IEEE Trans. Magn. 45, 155–159 (2009).
[Crossref]

Mali, R. K.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[Crossref]

Mescher, M. J.

M. J. Mescher, M. L. Vladimer, and J. J. Bernstein, “A novel high-speed piezoelectric deformable varifocal mirror for optical applications,” in “15th IEEE Int. Conf. Micro Electro Mech. Syst. (MEMS),” (IEEE, 2002), pp. 511–515.

Moreau, S.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Morrison, R. L.

Perreault, J.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[Crossref]

Pouydebasque, A.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
[Crossref]

Saint-Patrice, D.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Sinquin, J.

R. Gasmi, J. Sinquin, P. Jagourel, J. Dournaux, D. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future e-elt,” Proc. SPIE 7017, 70171Z (2008).
[Crossref]

Soulimane, S.

A. Pouydebasque, S. Bolis, F. Jacquet, C. Bridoux, L. Zavattoni, S. Soulimane, S. Moreau, D. Saint-Patrice, C. Bouvier, C. Kopp, and S. Fanget, “Thin varifocal liquid lenses actuated below 10v for mobile phone cameras,” Proc. SPIE 8252, 82520P (2012).
[Crossref]

Stack, R. A.

Steinich, T.

T. Steinich and V. Blahnik, “Optical design of camera optics for mobile phones,” Adv. Opt. Technol. 1, 51–58 (2012).

Stürmer, M.

B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

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.

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.

Suzuki, T.

I. Kanno, T. Kunisawa, T. Suzuki, and H. Kotera, “Development of deformable mirror composed of piezoelectric thin films for adaptive optics,” IEEE J. Sel. Top. Quantum Electron. 13, 155–161 (2007).
[Crossref]

Tremblay, E. J.

Tyholdt, F.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Ulvensøen, J. H.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Vladimer, M. L.

M. J. Mescher, M. L. Vladimer, and J. J. Bernstein, “A novel high-speed piezoelectric deformable varifocal mirror for optical applications,” in “15th IEEE Int. Conf. Micro Electro Mech. Syst. (MEMS),” (IEEE, 2002), pp. 511–515.

Wallrabe, U.

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.

B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

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.

Wang, D. T.

L. Henriksen, M. Eliassen, V. Kartashov, J. H. Ulvensøen, I.-R. Johansen, K. H. Haugholt, D. T. Wang, F. Tyholdt, and W. Booij, “Compact adjustable lens,” (Oct.25, 2011). US Patent8,045,280.

Wapler, M.

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.

Wapler, M. C.

B. P. Bruno, A. R. Fahmy, M. Stürmer, U. Wallrabe, and M. C. Wapler, “Properties of piezoceramic materials in high electric field actuator applications,” in preparation.

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.

Weirich, C.

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.

Zavattoni, L.

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Figures (13)

Fig. 1
Fig. 1 Schematic drawing of the buckling (left) and bending deformation (right).
Fig. 2
Fig. 2 Left: Drawing and simulation result of the mirror membrane at ( 1.2 , 1.2 ) kV mm, displacement scaled by a factor of XXX. Right: Closeup of the meshing at the contact of the glass membrane (blue shading), piezo rings (red) and polyurethane suspension (gray).
Fig. 3
Fig. 3 Left: Displacement of the mirror membrane for Rin = 11mm, hPZT = 150µm and different glass membrane thicknesses in the bending ( ( 0.3 , 0.3 ) kV mm ) and buckling ( ( 1.2 , 1.2 ) kV mm ) mode. Right: Local curvature (second radial derivative). The inset shows the buckling mode at large radii and the circles indicate R0; the 100 µm membrane had R0 > 0 in the bending mode.
Fig. 4
Fig. 4 Schematic drawing of the mirror membrane (left) and the full mirror system (right).
Fig. 5
Fig. 5 Illustration of some fabrication steps: Glueing of the membrane in two steps using vacuum chucks (a), fabrication of the PDMS mold (red) of the elastomer ring using a positive master (b), casting the polyurethane elastomer ring (c) and final assembly (d).
Fig. 6
Fig. 6 Focal power as a function of the applied electric field at the upper actuator in the bending mode (left, Elow = −Eup) and buckling mode (right, Elow = Eup). The measurements in the buckling mode include the positive and negative branch in a periodic buckling-only actuation and the decreasing electric field in the maximal trajectory.
Fig. 7
Fig. 7 Quartic coefficient in the reflected wavefront as a funtion of the focal power for the maximal trajectory described in the text. The top and right axes indicate the F-number (assuming an aperture of 9 mm) and the aspherical Zernike coefficient normalized to the d-line.
Fig. 8
Fig. 8 Left: Focal power (contour lines) and aspherical coefficient (colors) as a function of the applied voltage. Right: Linear shift of the reflected wavefront at its center relative to ( 0.3 , 0.3 ) kV mm (contour lines) and RMS wavefront error (colors).
Fig. 9
Fig. 9 Left: Frequency spectum (RMS displacement) of a ± 0.033 kV mm bending signal without bias and 0.67 kV mm buckling bias, normalized to the response without bias at 10 Hz. Right: Response to a rising (solid) and falling (dashed) 0.67 kV mm step applied to one piezo (mixed bending/buckling), normalized to the displacement at 0.5 s. The times indicate the rise/drop time of the linear rising and falling edge.
Fig. 10
Fig. 10 (a) From left to right: Polymer lens in the embossing mount with PDMS support (red), with glass lens as negative mold and finished lens. (b) Objective structure with fixed lens and primary mirror, complete system without primary aperture.
Fig. 11
Fig. 11 Left: Ray tracing diagram of the imaging system. Right: Spot diagrams with three different wavelengths in the center, at 1 mm radius (upper edge of the image) and 2 mm radius (far corner of the sensor). The grid lines are in steps of 20 µm.
Fig. 12
Fig. 12 Left: Focal power of the mirror for different image planes and the thin lens fit described in the text. The measurement data includes both the effective focal power feff (curvature and shift) and the contribution of the curvature of the mirror. Right: Measured and simulated resolution of the imaging system. The line indicates the resolution at 1400 mm scaled by the distance assuming constant magnification.
Fig. 13
Fig. 13 Left: Resolution test chart and image taken at 3 m distance (cropped to 52% and scaled in brightness) Right: Images of a watch at 0.59 m distance and a roof with a clock tower at 110m distance obtained at the two different foci. Both objects are present in both images.

Tables (1)

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Table 1 Summary of the simulation: The focal power and aspherical coefficient of different membrane designs in the bending mode (0.3,−0.3) kV/mm, combined bending and buckling (1.2,−0.3) kV/mm and buckling mode at low fields (0.3,0.3) kV/mm and high fields, (1.2,1.2) kV/mm. The usable aperture R0 is only shown for the pure bending and high-field buckling modes. The numbers are colored on a rainbow scale from red (lowest) to blue (highest).

Equations (7)

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Δ R = R d 31 1 2 ( E low + E up )
ρ curv . 1 ~ R 1 d 31 3 ( E low + E up ) .
r z membrane ~ R h d 31 ( E low E up ) .
ρ curv . 1 ~ h 1 d 31 ( E low E up ) .
z α 0 + 1 2 ( , 1 2 f 1 r 2 + α 4 r 4 + α 6 r 6 + + α 12 r 12 ) ,
Δ f 1 = ϕ Δ l opt , ϕ 14.00 × 10 3 m 2
L 1 ~ ρ 1 f eff 1 + L 0 1 .

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