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Abstract
We present a compact adaptive glass membrane lens for higher order wavefront correction and axial scanning, driven by integrated segmented piezoelectric actuators. The membrane can be deformed in a combination of rotational symmetry providing focus control of up to ± 6 m−1 and spherical aberration correction of up to 5 wavelengths and different discrete symmetries to correct higher order aberrations such as astigmatism, coma and trefoil by up to 10 wavelengths. Our design provides a large clear aperture of 12 mm at an outer diameter of the actuator of 18 mm, a thickness of 2 mm and a response time of less than 2 ms.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Wavefront distortions originating either directly from the optical elements or from the transmission medium affect the imaging quality by distorting the focal spot or destroying the coherence or shape of the wavefront [1]. While the distortions from the optical elements can be minimized through suitable engineering, the distortions from the transmission medium, such as the atmosphere, nutrition media or the specimen itself need active correction techniques.
In telescopes, adaptive optics use deformable mirrors to correct the incoming distorted wavefronts in real-time and improve the imaging resolution [2,3]. The technology has also been adapted for microscopy [4–6] to correct the aberrations induced by the sample, i.e., the distortions from the spatial variation of the refractive index in the sample or its surface. Deformable mirrors can be actuated by different principles like electrostatic with cantilever actuators as in [7–10], magnetic actuators using permanent magnets and coils as in [11–13]. The most traditional principle is piezoelectric, in some, a thin film of piezoceramic material with segmented electrodes is directly deposited or glued below the reflective surface to achieve a unimorph bending deformation and, in others, an array of stack actuators with thickness actuation mode are used to deform the reflective surface [14–24].
The reflective approach results in a folded beam path and increases the size of the imaging system, so, transmissive adaptive elements as in [25–32] can help to create more compact systems. A very interesting approach is the electrostatic deformation of a fluid-membrane interface using transparent electrodes in [28]. Most similar to our approach are the piezo-driven lenses of [31,32]. [31] uses two piezo-glass unimorph actuators, each equipped with $8$ electrode segments and the space in between the actuators is filled with a transparent liquid, a mineral oil, while one actuator corrects the defocus and astigmatism, the other corrects the coma and secondary astigmatism. [32] uses a single piezo-glass unimorph actuator with 32 electrode segments corrects comas, trefoils, astigmatisms and trefoils. All have in common that the ratio of aperture and outer diameter is below 0.5 (10 mm / 33 mm [28], 10 mm / 25 mm [31], 20 mm / 50 mm [32]) and that they have a small range of the focal power.
Applications like scanning microscopes [33,34] however, require dedicated tunable lenses with large focusing range and high speed, making the system again more complex. To reduce this complexity, the authors of [35] developed a lens with active focusing and passive spherical correction and we designed a type of lens (referred to as ‘base lens’) that allows for simultaneous focusing and active spherical aberration correction [36–38]. We successfully demonstrated applications of this lens for axial scanning and spherical aberration correction in confocal microscopy to improve the resolution [39,40] and recently modified it to control chromatic aberrations in a tunable achromatic configuration [41]. This lens uses a thin glass membrane as an optical surface, which results in an exceptional robustness against gravity and vibration, with a gravity-induced coma in vertical orientation of less than $\lambda /10$ [42]. In [43], we also found that the repeated operation of the piezo material in combination with a passive glass layer and the glue that we use in this paper is stable in the long term. Furthermore, the compact version of the base lens was reliably tested over $8$ million cycles [38].
To go one step further, we propose an alternative approach to address both a large focal shift for axial scanning and also higher order wavefront correction with a single component based on our highly responsive composite piezo-glass sandwich membrane of [36–38]. We modified this membrane by dividing the electrodes on the two piezo layers into annular segments to allow for asymmetric deformations of the optical surface and thus achieve higher order wavefront corrections while still retaining the large tunable focus range.
In comparison to the state of the art, our lens is highly integrated with minimal mechanical complexity, requiring only the minimum $8$ control signals, compared to the dual-membrane, dual actuator design of [27], the 32 control signals of [32] or the complex actuator layout of [26]. It is more compact, either laterally or longitudinally than most of the literature [26,27,32], in addition to a larger range of the focal power of $\pm$ 6 m$^{-1}$ and a short response time of less than 2 ms. Also some of the literature lacks some of the spatial degrees of freedom, as [29,32] cannot control the spherical aberration and [29] is lacking trefoil. In addition, our element can be readily converted into an achromat according to [41] and also miniaturized further [38].
In this paper, we introduce two possible layouts of the active membrane for higher order correction: 4 radial segments on each of the two piezo layers with a ${45}^{\circ }$ overlap between them and an $8$ radial segment on only one of the piezo layers. We describe their design and fabrication in section 2. In section 3, we present the characterization of the adaptive lens with the wavefront correction range, axial scanning range and speed, and we conclude the paper in section 4.
2. Design and fabrication
The adaptive lens is based on a deformable ultra-thin glass membrane, that is sandwiched between two piezoceramic rings to form an active optical surface, and combined with a deformable fluid chamber and an optical fluid, to form the lens body [36,37], as in Fig. 1(a).
Fig. 1. (a.) Schematic cross section of the adaptive lens in the bending (b.) and buckling (c.) actuation modes, and the segmentation of electrodes on the two piezo layers and images of the adaptive lenses in version A (d.) and version B (e.). The outer dimensions of the PCB are designed to be compatible with 30 mm pitch system [44] for best handling during the characterization.
Upon application of electric fields to the piezo layers, the membrane can be deformed to a convex or a concave shape, and the fluid underneath adapts to the membrane surface to essentially form a plano-convex or a plano-concave lens.
With two degrees of freedom in the application of electric fields on the two piezos, we can achieve two deformation modes. First, a bending mode by applying opposite electric fields on the two piezos to expand or contract them with different magnitudes and achieve a bending moment to deform the membrane with a rather spherical or elliptical profile, as in Fig. 1(b). Second, a buckling mode by applying equal electric fields on the two piezos to contract both the piezos radially and achieve a compressive force to deform the membrane through buckling with a more hyperbollical profile, as in Fig. 1(c). We can also combine the two modes to simultaneously tune the curvature and the asphericity, essentially the focal power and the spherical aberration by giving the glass membrane tunable Dirichlet and von Neumann boundary conditions [36–38].
Rather than using uniform electrodes on the piezo layers, which give a rotationally symmetric deformation of the membrane, we now divide the electrodes into annular segments, which allow individual actuation of the segments and hence an asymmetric deformation of the membrane. With different combinations of actuated segments, we aim to correct higher-order wavefront aberrations. In version $A$, we divided four segments on both the top and bottom piezo layers and aligned them to have a ${45}^{\circ }$ overlap between them, as shown in Fig. 1(d) and in version $B$, we divided only the top piezo layer into $8$ annular segments, as in Fig. 1(e), which may be less elegant and adds an additional degree of freedom, but allows for more straightforward contacting and avoids rotational alignment errors.
We used a nanosecond UV laser (Trumpf TruMark $6330$) for structuring the components, the glass membrane and the glass window (50 µm and 500 µm thick from SCHOTT $D263t$ eco [45]), a generic FR-$4$ substrate (500 µm thick), and the piezoceramic films (M$1100$: 100 µm thick from Johnson Matthey [46]). Using the same laser, we also structured the outer electrode layer on the piezos to create the annular segments.
We glued the piezo rings on either side of the glass membrane using a high-temperature epoxy glue [47] and used alignment markers to fix the overlap of the segments of version $A$. We then cast the fluid chamber from polyurethane (Smooth On ClearFlex-$50$ [48]) and sequentially glued the sandwich actuator, fluid chamber, and the FR-$4$ substrate with the glass window. To finish, we filled the chamber with paraffin oil ($n_{D\!,20}=1.47$) [49] using a pressurized fluid dispenser and a vacuum arrangement, and contacted the segments with 50 µm thick insulated copper wires by soldering. Photos of the fabricated versions of the adaptive lenses are shown in Fig. 1(d and e).
3. Opto-mechanical characterization
In this paper, we obtain the relevant optical information, i.e., the wavefront shift, from the surface profile as this can be measured very reliably with a high axial and lateral resolution over a large measurement range. As we can expect that the ${500}\;\mathrm{\mu}\textrm{m}$ glass substrate on the back side of the lens remains flat and the fluid remains homogeneous (at relatively low frequencies), the full optical effect comes from the surface and deformation of the glass membrane. The membrane itself is just ${50}\;\mathrm{\mu}\textrm{m}$ thick, so it is much thinner than the curvature radius (80 mm at maximum deformation) and its refractive index is close to the fluid ($1.52$ vs. $1.47$). Hence, the ’thick lens’ effects of the glass membrane are negligible, as is the opto-mechanical effect. In [38], we calculated a worst-case distortion of less than ${1.5}\;\textrm{nm}$ at maximum focal power in a similar lens, well below the accuracy of our measurements. Similarly, the surface roughness of the membrane material is less than ${1}\;\textrm{nm}$ [45]. Moreover, the characterization of the base lens with surface method in [37] and optical methods in [50] were satisfactorily comparable.
To measure the surface profile, we used a translational stage with a step size of ${200}\;\mathrm{\mu}\textrm{m}$ and a confocal distance sensor providing a resolution of vertically ${2.7}\;\textrm{nm}$ with a $10$x averaging. To eliminate the edge effects between the piezo and the glass membrane due to fabrication imperfections, we considered only an aperture of 10 mm, ignored any noise data points due to dust particles and also subtracted the surface deformation of the lens in the neutral state as shown in Fig. 2 for the computation of the aspherical profile and the Zernike coefficients. For the former, we used a $4^{\textrm {th}}$ order rotationally symmetric fit with a least square regression method and for the latter, we used the ZernikeCalc [51] function in Matlab. The $R^2$ of this fit varies depending on the displacement, for the pre-displacement, we had $R^2= 0.86$, for the asymmetric modes in section 3.1, we had between $0.9$ and $0.93$ and for the largest symmetric deformations in the buckling mode 3.1, we had above $0.98$.
Fig. 2. Lens profile in the neutral state showing a slight pre-deflection (a), corresponding Zernike coefficients (b) and remaining residuals (c).
While the pre-deflection in Fig. 2 appears large, we will see that it is small compared to the tuning range of the corresponding Zernike coefficients, so it can be easily removed actively. Furthermore, we expect that it would be significantly smaller in an optimized industrial fabrication process. In particular, the defocus highly depends on the filling process. In the residuals map, as shown in Fig. 2(c), with an RMS residual of ${20.73}\;\textrm{nm}$, we see that Zernike fit captures the surface profile sufficiently well.
We actuated the lenses in two conditions: non-uniform electric fields applied on the annular segments of each piezo layer to achieve an asymmetric deformation of the membrane for higher order wavefront correction and uniform electric fields applied on all the segments of each piezo layer to achieve a rotationally symmetric deformation of the membrane for axial scanning. The complete set of electric field is shown in Fig. 3.
Fig. 3. Electric field combinations (a. version $A$: $A_1$ to $A_8$; b. version $B$: $B_1$ to $B_9$) applied on the adaptive lenses to achieve rotationally symmetric deformations of different symmetries.
While we show the rotationally symmetric deformations at large fields and with the non-linear buckling effect, we restrict ourselves for the asymmetric deformations to the linear regime near a flat membrane to demonstrate the mechanical degrees of freedom.
3.1 Higher order wavefront correction
In this section, we should keep in mind that we see only a small part of the picture as we only consider perturbations in bending modes, i.e., with net vanishing compression, around the flat state of the lens. Those results will certainly depend on the overall pre-deflection of the lens, and there may be also non-linear buckling-type deformations, but those will be too lengthy to discuss here. This reduces the degrees of freedom by one, i.e version $A$ with $8$ segments has $7$ degrees of freedom and version $B$ with $9$ total segments has $8$ degrees of freedom and as a consequence excludes controlling the spherical Zernike coefficient. This also translates to number of aberrations that we can correct - for instance, $2$ x astigmatism, $2$ x coma, $2$ x trefoil, defocus, and quadrafoil (version $B$).
We actuated the adaptive lenses with the different combinations of the electric fields intuitively defined in Fig. 3 to address possible higher order aberrations; limited to $\pm {0.4}\;\textrm{kV}\;\textrm{mm}^{-1}$ to confine the piezos deformation to a linear regime and avoid depolarization. We then fitted the membrane surface with Zernike polynomials ($Z$) by the definitions in [52] using the ZernikeCalc [51] function in Matlab and computed the wavefront modulation ($W$) in terms of Zernike coefficients ($C$):
with all angular frequencies $m$ up to radial order $n=4$.The astigmatism can be addressed directly with the combinations $A_5$, $A_6$, $B_5$ and $B_6$ as in Fig. 3(a, b) with the corresponding response is shown in Fig. 4(a, b). Furthermore, in version $B$, we can additionally address directly one quadrafoil with the combination $B_9$, as shown in Fig. 4(c). The comas and trefoils cannot be directly addressed by the modes in Fig. 3, instead the semi ($A_1$, $A_2$, $B_1$ and $B_2$) and quarter modes ($A_3$, $A_4$ or $B_3$, $B_4$) create a combination of tilt, coma and trefoil as shown in Fig. 5(a to d). Hence, we define the coupling matrix
Fig. 4. Displacement of the ${90}^{\circ }$ anti-symmetric modes (a, b) and the ${45}^{\circ }$ anti-symmetric mode (c.) at $\pm {0.4}\;\textrm{kV}\;\textrm{mm}^{-1}$ in terms of Zernike coefficients. The data was measured for version $B$ and is very similar to version $A$. (O: Oblique; V: Vertical; H: Horizontal; Ast: Astigmatism; Quad: Quadrafoil)
Fig. 5. Coupling of (a.) semi $A_1$, (b.) semi $A_2$, (c.) quarter $A_3$ and (d.) quarter $A_4$ modes with tilts, comas and trefoils at $\pm {0.4}\;\textrm{kV}\;\textrm{mm}^{-1}$. The data was measured for version $A$ and is very similar to version $B$. (O: Oblique; V: Vertical; H: Horizontal; Ast: Astigmatism; Quad: Quadrafoil)
Version $B$ is analogous to version $A$. In Fig. 6, we show the possibility to address the vertical coma independently by adding combinations $A_2$ and $A_4$ with the ratio $\frac {-0.039}{0.055}$ and vertical trefoil using the ratio $\frac {-0.037}{-0.052}$ as computed from the matrix Eq. (3). Any remaining tilts can be corrected by employing, for example, our adaptive prisms, which can correct tilts, e.g., by up to 0.6 mm [53] or for static imaging applications by digitally correcting the image. While the maximum coma of ${0.7}\;\mathrm{\mu}\textrm{m}$ is the smallest of all the tuning ranges of the Zernike coefficients, it is approx $10$ times larger than the gravitational coma observed in [42], so the latter can be corrected with just ${0.04}\;\textrm{kV}\;\textrm{mm}^{-1}$.
Fig. 6. Superposition of the semi $A_2$ and quarter $A_4$ modes at maximum fields of $\pm {0.4}\;\textrm{kV}\;\textrm{mm}^{-1}$ for correcting vertical (a.) coma and (b.) trefoil. The data was measured for version $A$ and is very similar to version $B$. (O: Oblique; V: Vertical; H: Horizontal; Ast: Astigmatism; Quad: Quadrafoil)
For reasons of compactness, we have included the full coupling matrices of all Zernike polynomials in the Supplement 1 Eq. S$1$ to Eq. S$4$.
We show all the other unintended Zernike coefficients that arise during actuation and were not visible in the other plots in Fig. 7. They contain on the one hand imperfections of the fabrication process, the geometry of the electrodes and in the case of the composed trefoil and coma also the non-linearity of the piezo. On the other hand, they also include effects from the cross-talk between electrodes. However, considering cross-talk with different modes, for instance the ratio between trefoil to coma in the coma mode, or coma to trefoil in the trefoil mode are nearly zero (${0.007}\;\%$ and ${0.009}\;\%$, respectively). To investigate potential cross-talk further, we also simulated the electric field distribution using COMSOL Multiphysics when applying opposite electric potentials to adjacent segments as shown in Fig. 8(a). We see that the field is homogenous and vertical up to close to the insulation groove. In the insulation region, there is a significant horizontal component, the mechanical results of which are hard to predict as they depend on possible re-polings due to the operation history of the lens. Their effect is, however, relatively small, as the region is just ${100}\;\mathrm{\mu}\textrm{m}$ wide, with an overall length of a single electrode of approx. 12 mm.
Fig. 8. a. Simulation of electric field distribution (arrows) and potential (colors) in the cross section of the insulation region between neighboring electrodes. b. Non-linear behavior of the lens in different modes due to piezo hysteresis.
The behavior of the lens will inherit the hysteresis and non-linearity from the piezo, as we show in Fig. 8(b). This is, however, not relevant in closed loop operations, such as using neural networks without additional sensors [54,55] or when operating the lenses on defined voltage trajectories.
So far, we have demonstrated the correction of astigmatism, coma, trefoil and also one quadrafoil for version $B$, and in the next section, we will show the range of defocus and spherical aberration.
3.2 Axial scanning
In this section, we determine the axial scanning (focal power) and spherical correction limit with the rotationally symmetric modes, i.e combinations $A_7$, $A_8$, $B_7$ and $B_8$ in Fig. 3. We fitted the measured membrane profile over a radius $r_m$ with a $4^{\textrm {th}}$ order rotational symmetric fit, defined by
In Fig. 9, we show the axial scanning response of the adaptive lenses in the rotationally symmetric modes. We see the hysteresis behavior of the piezo material with a small, and symmetric focal power range of $\pm$0.3 m−1 in Fig. 9(a) for the symmetric electric fields with an amplitude of $\pm$0.4 kV mm−1 in the bending mode $A_7$ or $B_7$. The graph appears noisy due to small displacement compared to the buckling mode. The pure buckling mode $A_8$ or $B_8$ with fields up to 1.5 kV mm−1, in contrast, results in a highly non-linear response and a uni-directional focal power of 6.5 m−1, as shown in Fig. 9(b). We also observe a large pre-deflection at ${0}\;\textrm{kV}\;\textrm{mm}^{-1}$ in the buckling mode, probably due to hysteresis and creep of the piezos as we periodically applied only positive electric fields. Furthermore, we actuated the lens in a combination of bending and buckling to outline the tunable range, i.e., a range from a parabolic profile of the bending mode to a hyperbolic profile from the buckling mode shown in Fig. 9(c), for the electric field trajectory of Fig. 9(f). Both the versions have a similar tunable range that is similar to the base lens in [36,37], with an aspherical tuning rang of approx. ${3}\;\mathrm{\mu}\textrm{m}$ over most of the focal range of $\pm$6.5 m−1. The deviations between both versions are of the order of typical variations between prototypes in the fabrication process, which is very sensitive in our experience.
Fig. 9. Defocus ($f^{-1}= 2(n_{D\!,20}-1)\alpha _2$) as a function of the applied electric fields for the bending mode (a., d.) and the buckling mode (b., e.); outline of the operating region of defocus and aspherical correction in a combination of bending and buckling (c., f.) with relevant points of the field combinations highlighted.
Fig. 10. Lens profile in the maximum buckling mode (a) with corresponding residuals (b), Zernike coefficients for maximum buckling (c) and bending (d), and unintended aberrations in different modes (e.).
To demonstrate the largest possible aberrations that may occur at large focal powers, we show the surface fit, residual and aberration at a maximum focal power of ${6.5}\;\textrm{m}^{-1}$ in Fig. 10. In particular, the residual with RMS of $3.69$ nm is better than un-actuated state residual with RMS of $20.17$ nm in Fig. 2, due to the stronger stress in the membrane that leads to a more stable surface. The remaining aberrations are in the tuning range of the corresponding modes.
3.3 Response time
We have determined the step response of the adaptive lenses by actuating them with a ${1}\;\textrm{Hz}$ step function of $\pm {0.4}\;\textrm{kV}\;\textrm{mm}^{-1}$ in the semi mode ($A_1$), astigmatism mode ($A_5$) and bending mode ($A_7$) and measured the deformation at a single point on the membrane. The responses as shown in Fig. 11 show a rise time well below a millisecond and a slower fall time of roughly ${2}\;\textrm{ms}$ along with a settling time of few milliseconds, which can be suppressed by a closed-loop control or significantly reduced with a smoother excitation [56]. Hence, we assume that with an active control, one can achieve a response below a millisecond. We notice on the one hand, that the bending mode and astigmatism have approximately same response, probably as defocus and astigmatism are both of $2^{\textrm {nd}}$ order radially. We see the resonances better in Fig. 11(c), where we find resonances of ${685}\;\textrm{Hz}$ for the astigmatism and ${1083}\;\textrm{Hz}$ for the bending mode. This agrees reasonably well with the ratio of $1.67$ for the ($2,1$) and ($0,1$) modes of the resonance of a disk found in [57]. With quality factors of $2.28$ and $3.03$, the astigmatism has higher damping than the defocus, in agreement with the results of the step response, probably as the astigmatism creates a greater deformation of the edge of the membrane, where the plasticity of the polyurethane causes damping. This interpretation is in agreement with the very highly damped semi-vertical mode with a quality factor of $1.06$, which is probably predominantly an overall tilt of the membrane. This analysis further agrees with the findings in [38] that fluids with different viscosities do not significantly influence the response time.
Fig. 11. Rising (a), falling (b) and frequency (c) response of the adaptive lens (version $A$) in different actuation modes. The inset shows the measurement point on the membrane.
4. Summary
We have presented adaptive lenses with higher order wavefront correction and a large axial scanning range. The lenses are based on an active membrane using a piezo-glass-piezo sandwich configuration that leads to a compact and integrated layout [36] and an exceptional robustness against gravity and vibration, with a gravity-induced coma in vertical orientation of less than $\lambda /10$ [42]. To excite the full asymmetric Zernike coefficients up to $3^{\textrm {rd}}$ order (plus spherical), we divided the electrode on the piezoelectric layer into annular segments to allow for the necessary spatial degrees of freedom in the deformation of the membrane. In version $A$, we had $4$ segments on each layer, rotated by ${45}^{\circ }$, and in version $B$, we had $8$ segments on the top layer providing more straightforward fabrication, but adding a redundant degree of freedom. The lens has an aperture of 12 mm at a small actuator footprint of 18 mm, a thickness of only 2 mm, a response time of the order of 2 ms that can be potentially reduced well below a millisecond with active control and first order resonances at ${685}\;\textrm{Hz}$ (astigmatism) and ${1080}\;\textrm{Hz}$ (defocus).
We obtained the Zernike coefficient of the wavefront shift for a complete set of voltage combinations that cover all degrees of freedom with suitable symmetries by measuring the surface profiles of the lenses with a profilometer. This gave us both the maximum tuning range (around a neutral state) and a matrix that provides the electric field combinations for any required wavefront shift. The tunable focal power for axial scanning is around $\pm$ 6 m−1 and the maximum wave front shifts are around $\pm$ 5 µm for spherical aberrations, $\pm$ 5 µm for astigmatism, and $\pm$ 0.6 µm for the quadrafoil given conservative limitations to prevent piezo depolarization [43] and electrostatic breakdown. Furthermore any two of the three interdependent tilt, coma and trefoil, can be independently corrected, giving, a range of coma of up to $\pm$ 0.7 µm and trefoil of up to $\pm$ 0.7 µm. Also, both the aberrations in the neutral state and at maximum focal powers are well below the tuning range of the lens and any remaining residuals are at worst ${21}\;\textrm{nm}$ (RMS). Overall, we can control the focus and all aberrations up to $3^{\textrm {rd}}$ order plus spherical, any remaining tilts can be corrected using adaptive prisms [53] or in some situations digital image post-processing.
Furthermore, two of these adaptive lenses can be combined in an achromatic configuration as in [41] to provide both chromatic and geometric aberration correction in a single device. Our general concept can be also scaled to other sizes, such as the extremely compact lenses in [38].
In the future, we plan to employ the adaptive lens in confocal microscopy similar to [39,40] for imaging biological tissues in a closed loop control that will also avoid the issue of piezo non-linearity and hysteresis.
Funding
Deutsche Forschungsgemeinschaft (1657/6-2, 4555/2-1); Otto von Guericke University Magdeburg (Open Access Publication Fund).
Acknowledgments
We thank Dr. Florian Lemke and Ms. Yasmina Frey for earlier work on the segmentation of piezos.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper is not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. W. Charman, “Wavefront technology: Past, present and future,” Contact Lens and Anterior Eye 28(2), 75–92 (2005). [CrossRef]
2. J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31(1), 13–62 (1993). [CrossRef]
3. A. Rukosuev, A. Kudryashov, A. Lylova, V. Samarkin, and Y. V. Sheldakova, “Adaptive optics system for real-time wavefront correction,” Atmos. Oceanic Opt. 28(4), 381–386 (2015). [CrossRef]
4. M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. R. Soc., A 365(1861), 2829–2843 (2007). [CrossRef]
5. N. Ji, “Adaptive optical fluorescence microscopy,” Nat. Methods 14(4), 374–380 (2017). [CrossRef]
6. W. G. Jerome and R. L. Price, Basic confocal microscopy (Springer, 2018).
7. T. G. Bifano, R. Krishnamoorthy Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, “Continuous-membrane surface-micromachined silicon deformable mirror,” Opt. Eng. 36(5), 1354–1360 (1997). [CrossRef]
8. K. Bush, D. German, B. Klemme, A. Marrs, and M. Schoen, “Electrostatic membrane deformable mirror wavefront control systems: design and analysis,” in Advanced Wavefront Control: Methods, Devices, and Applications II, vol. 5553 (SPIE, 2004), pp. 28–38.
9. T. Bifano, P. Bierden, and J. Perreault, “Micromachined deformable mirrors for dynamic wavefront control,” in Advanced Wavefront Control: Methods, Devices, and Applications II, vol. 5553J. D. Gonglewski, M. T. Gruneisen, and M. K. Giles, eds., International Society for Optics and Photonics (SPIE, 2004), pp. 1–16.
10. S. Bonora and L. Poletto, “Push-pull membrane mirrors for adaptive optics,” Opt. Express 14(25), 11935–11944 (2006). [CrossRef]
11. O. Cugat, S. Basrour, C. Divoux, P. Mounaix, and G. Reyne, “Deformable magnetic mirror for adaptive optics: technological aspects,” Sens. Actuators, A 89(1-2), 1–9 (2001). [CrossRef]
12. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express 15(26), 18190–18199 (2007). [CrossRef]
13. R. Hamelinck, R. Ellenbroek, N. Rosielle, M. Steinbuch, M. Verhaegen, and N. Doelman, “Validation of a new adaptive deformable mirror concept,” in Adaptive optics systems, vol. 7015 (SPIE, 2008), pp. 163–174.
14. T. Sato, H. Ishida, and O. Ikeda, “Adaptive pvdf piezoelectric deformable mirror system,” Appl. Opt. 19(9), 1430–1434 (1980). [CrossRef]
15. Y. Hishinuma and E.-H. Yang, “Piezoelectric unimorph microactuator arrays for single-crystal silicon continuous-membrane deformable mirror,” J. Microelectromech. Syst. 15(2), 370–379 (2006). [CrossRef]
16. 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(2), 155–161 (2007). [CrossRef]
17. P. Yang, Y. Liu, W. Yang, M.-W. Ao, S.-J. Hu, B. Xu, and W.-H. Jiang, “Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror,” Opt. Commun. 278(2), 377–381 (2007). [CrossRef]
18. A. Norton, J. W. Evans, D. Gavel, D. Dillon, D. Palmer, B. Macintosh, K. Morzinski, and S. Cornelissen, “Preliminary characterization of boston micromachines’ 4096-actuator deformable mirror,” in MEMS adaptive optics III, vol. 7209 (SPIE, 2009), pp. 134–140.
19. S. Verpoort and U. Wittrock, “Actuator patterns for unimorph and bimorph deformable mirrors,” Appl. Opt. 49(31), G37–G46 (2010). [CrossRef]
20. K. L. Wlodarczyk, E. Bryce, N. Schwartz, M. Strachan, D. Hutson, R. R. J. Maier, D. Atkinson, S. Beard, T. Baillie, P. Parr-Burman, K. Kirk, and D. P. Hand, “Scalable stacked array piezoelectric deformable mirror for astronomy and laser processing applications,” Rev. Sci. Instrum. 85(2), 024502 (2014). [CrossRef]
21. H. Wang, “Research on a bimorph piezoelectric deformable mirror for adaptive optics in optical telescope,” Opt. Express 25(7), 8115–8122 (2017). [CrossRef]
22. Z. Zhu, Y. Li, J. Chen, J. Ma, and J. Chu, “Development of a unimorph deformable mirror with water cooling,” Opt. Express 25(24), 29916–29926 (2017). [CrossRef]
23. V. Toporovsky, A. Kudryashov, A. Skvortsov, A. Rukosuev, V. Samarkin, and I. Galaktionov, “State−of−the−art technologies in piezoelectric deformable mirror design,” in Photonics, vol. 9 (MDPI, 2022), p. 321.
24. X. Han, J. Ma, K. Bao, Y. Cui, and J. Chu, “Piezoelectric deformable mirror driven by unimorph actuator arrays on multi-spatial layers,” Opt. Express 31(8), 13374–13383 (2023). [CrossRef]
25. A. N. Simonov, G. Vdovin, and M. Loktev, “Liquid-crystal intraocular adaptive lens with wireless control,” Opt. Express 15(12), 7468–7478 (2007). [CrossRef]
26. P. Liebetraut, S. Petsch, J. Liebeskind, and H. Zappe, “Elastomeric lenses with tunable astigmatism,” Light: Sci. Appl. 2(9), e98 (2013). [CrossRef]
27. S. Bonora, Y. Jian, P. Zhang, A. Zam, E. N. Pugh, R. J. Zawadzki, and M. V. Sarunic, “Wavefront correction and high-resolution in vivo oct imaging with an objective integrated multi-actuator adaptive lens,” Opt. Express 23(17), 21931–21941 (2015). [CrossRef]
28. P. Rajaeipour, M. Sauther, K. Banerjee, H. Zappe, and Ç. Ataman, “Seventh-order wavefront modulation with a gravity-neutral optofluidic deformable phase plate,” J. Optical Microsystems 1(03), 034502 (2021). [CrossRef]
29. D. Kopp and H. Zappe, “Tubular astigmatism-tunable fluidic lens,” Opt. Lett. 41(12), 2735–2738 (2016). [CrossRef]
30. D. Kopp, Tubular optofluidics (Albert-Ludwigs-Universität Freiburg, 2017).
31. S. Bonora, S. Lee, Y. Jian, M. Cua, E. N. Pugh, R. J. Zawadzki, and M. V. Sarunic, “Multi-actuator adaptive lens for wavefront correction in optical coherence tomography and two-photon excitation fluorescence microscopy (conference presentation),” in Adaptive Optics and Wavefront control for biological systems II, vol. 9717 (SPIE, 2016), p. 971702.
32. T. Peng, C. Dai, J. Lou, Y. Cui, B. Tao, and J. Ma, “A low-cost deformable lens for correction of low-order aberrations,” Opt. Commun. 460, 125209 (2020). [CrossRef]
33. D. Shotton and N. White, “Confocal scanning microscopy: three-dimensional biological imaging,” Trends Biochem. Sci. 14(11), 435–439 (1989). [CrossRef]
34. D. M. Shotton, “Confocal scanning optical microscopy and its applications for biological specimens,” J. Cell Sci. 94(2), 175–206 (1989). [CrossRef]
35. P. Zhao, Ç. Ataman, and H. Zappe, “Spherical aberration free liquid-filled tunable lens with variable thickness membrane,” Opt. Express 23(16), 21264–21278 (2015). [CrossRef]
36. M. C. Wapler, M. Stürmer, and U. Wallrabe, “A compact, large-aperture tunable lens with adaptive spherical correction,” in 2014 International Symposium on Optomechatronic Technologies, (IEEE, 2014), pp. 130–133.
37. 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]
38. M. C. Wapler, “Ultra-fast, high-quality and highly compact varifocal lens with spherical aberration correction and low power consumption,” Opt. Express 28(4), 4973–4987 (2020). [CrossRef]
39. K. Philipp, A. Smolarski, N. Koukourakis, A. Fischer, M. Stürmer, U. Wallrabe, and J. W. Czarske, “Volumetric hilo microscopy employing an electrically tunable lens,” Opt. Express 24(13), 15029–15041 (2016). [CrossRef]
40. N. Koukourakis, M. Finkeldey, M. Stürmer, C. Leithold, N. C. Gerhardt, M. R. Hofmann, U. Wallrabe, J. W. Czarske, and A. Fischer, “Axial scanning in confocal microscopy employing adaptive lenses (cal),” Opt. Express 22(5), 6025–6039 (2014). [CrossRef]
41. H. G. Gowda, M. C. Wapler, and U. Wallrabe, “Tunable doublets: piezoelectric glass membrane lenses with an achromatic and spherical aberration control,” Opt. Express 30(26), 46528–46540 (2022). [CrossRef]
42. H. G. Gowda, B. P. Bruno, M. C. Wapler, and U. Wallrabe, “Reliability of tunable lenses: feedback sensors and the influence of temperature, orientation, and vibrations,” Appl. Opt. 62(12), 3072–3082 (2023). [CrossRef]
43. 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]
44. A. Afshari, “Optical bench system,” (1998). US Patent 5,828,502.
45. “schott-d-263-t-eco-thin-glass-may-2013-eng.pdf,” https://www.schott.com/d/advanced_optics/ea165ca6-b4f8-49ee-b989-072ddb2d2e4d/1.0/schott-d-263-t-eco-thin-glass-may-2013-eng.pdf. (Accessed on 09/28/2021).
46. “Dokument 8.1.2 - piezoceramic masses - data sheet englisch v2.00.0004,” https://www.piezoproducts.com/wp-content/uploads/2020/03/JM_Piezo_Products-Data-Sheet-Piezoceramic-Masses-v2.00.0004.pdf. (Accessed on 09/28/2021).
47. “Resoltech htg 240. high tg structural infusion epoxy systems,” https://www.resoltech.com/en/markets/htg240-detail.html. (Accessed on 02/19/2022).
48. “Clear flex 50 product information | smooth-on, inc.,” https://www.smooth-on.com/products/clear-flex-50/. (Accessed on 09/28/2021).
49. “Paraffin oil suitable for ir spectroscopy | 8012-95-1,” https://www.sigmaaldrich.com/DE/en/product/sial/76235?context=product. (Accessed on 11/29/2021).
50. 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]
51. Robert Gray, “Zernikecalc - file exchange - MATLAB central,”MathWorks (2013) [accessed 28 Sept 2021], https://de.mathworks.com/matlabcentral/fileexchange/33330-zernikecalc.
52. V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011). [CrossRef]
53. F. Lemke, P. M. Weber, K. Philipp, J. W. Czarske, N. Koukourakis, U. Wallrabe, and M. C. Wapler, “Piezo-actuated adaptive prisms for continuously adjustable bi-axial scanning,” Smart Mater. Struct. 29(9), 095004 (2020). [CrossRef]
54. K. Schmidt, N. Koukourakis, and J. W. Czarske, “Assignment of focus position with convolutional neural networks in adaptive lens based axial scanning for confocal microscopy,” Appl. Sci. 12(2), 661 (2022). [CrossRef]
55. K. Schmidt, N. Guo, W. Wang, J. Czarske, and N. Koukourakis, “Chromatic aberration correction employing reinforcement learning,” Opt. Express 31(10), 16133–16147 (2023). [CrossRef]
56. 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]
57. P. Leclaire, J. Goossens, L. Martinez, N. Wilkie-Chancelier, S. Serfaty, and C. Glorieux, “Study of the bending modes in circular quartz resonators,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 53(10), 1934–1943 (2006). [CrossRef]
