Modeling and design of piezoelectrically actuated MEMS tunable lenses
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Autofocus is a crucial feature in cameras, especially when photographing objects at different distances and having them in sharp focus without any quality loss in the captured image. Over the last decade, several research efforts have been made to incorporate tunable focus for mobile-device cameras using micro-scale components. Qualitatively, this would enable miniaturized cameras with lower power consumption, much faster response in scanning focus range and higher reliability. The microelectromechanical-systems-(MEMS)-based tunable focus lenses are promising alternatives as autofocus mechanisms when compared to the conventional macro-scale approaches such as the Voice Coil Motor (VCM)  or ultrasonic motor . Moreover, such MEMS autofocus lenses would achieve higher resolution smartphone cameras without having any moving parts within the camera housing, which consumes power during focus adjustment and causes a loss in the Field-of-View (FoV) as for the VCM. The research reported in this thesis is to construct a modeling framework for the piezoelectrically actuated MEMS tunable lenses on the electromechanical domain by finding an approximation for the lens displacement, and using it afterwards in the optical domain to find the lens' optical performance. Given the modeling framework, two design concepts have been proposed. The first one is to achieve larger lens apertures while having a tradeoff between focal length and RMS-wavefront error (RMSWFE), while the second is to increase lens' tunable range of focal lengths by controlling layers' stresses during fabrication. To approximate the lens displacement, we have used Hamilton's principle to deduce a variational formulation that can be easily solved in MATLAB . This has resulted in taking less calculation time than the time is taken by finite element method (FEM) programs such as COMSOL . The proposed displacement ansatz (weighted Gegenbauer polynomials) to approximate the lens displacement, has been chosen because they can be mathematically expressed in terms of Zernike polynomials. Those polynomials are suitable for representing the lens' wavefront when it comes to optical performance, which allows an exact mapping of the lens displacement profile to optical programs (e.g. Zemax ). Without this proposed framework, lens designers would have to use FEM simulations and over-mesh the pupil area before exporting the lens sag to optical programs; which is timeconsuming with dense meshing. Less calculation time, with our modeling framework, for the lens displacement originates from the pre-calculations of (linear and nonlinear) variational integrals in terms of the actuator's geometrical parameters. This has enabled storing mathematical expressions for the variational integrals that can be called once needed. For a new actuator's geometrical parameter, we can use a simple substitution to calculate the new displacement profile. Chapter 3 describes the proposed modeling framework for these type of lenses. We have considered different polygonal pupil geometries to explore if a design tradeoff can be gained in the optical performance. We have found out the first design concept called as pupil masking. With a 45-rotated square opening in the piezoelectric actuator, while keeping the lens pupil circular, it gives a tradeoff between the lens' optical parameters,e.g., lower RMSWFE at the expense of having larger focal length f allowing having large lens apertures. The proposed modeling framework (in Ch. 3) has a weakness that it has not accounted for the discontinuity of the lens layered structure around the pupil boundary. This required an increase of the model's degrees of freedom upto 120 in order to converge to a solution with a decent accuracy. Thus, in chapter 4, we have proposed having two new ansätze that use the aforementioned weighted Gegenbauer polynomials and, in addition, the exact solutions of the circular plate's differential equations. This has improved the speed of convergence to a solution and enabled having reduced-order models, which provide systemlevel designers with computationally efficient models. Yet, the new ansätze can be mapped to Zernike polynomials as well. Chapters 3 and 4 have dealt with the linear performance that is less accurate in case of large actuation voltages. The linear model also neglects residual stresses resulting from fabrication. Thus, we have proposed in chapter 5, to use von Kármán's plate theory instead of Kirchhoff theory. As a result, we have been able to consider the effect of having different residual stresses within the lens' layered structure and larger actuation voltages. Through the understanding of the model parameters, we have been able to propose the second design concept. By controlling the residual stresses during fabrication, the lens' tunable range of focal lengths can be increased by having the lens operating, depending on the driving voltage, as a plano-convex or a plano-concave lens. The proposed modeling frameworks have been verified versus FEM simulation as a reference point and moreover the nonlinear model has been verified versus measurements as well. In practice, these developed models can be utilized for optimization of different material choices and layers thicknesses to find the optimum geometrical parameter of the piezoelectric actuator. Finally, we provide conclusions and proposals for future work to build a dynamic model for the lens.
Består avArticle 1: Mahmoud A. Farghaly, Muhammad Nadeem Akram, and Einar Halvorsen, "Modeling framework for piezoelectrically actuated MEMS tunable lenses," Opt. Express 24, 28889-28904 (2016).
Aticle 2: Mahmoud A. Farghaly, Muhammad Nadeem Akram, Einar Halvorsen, "Optical performance of piezoelectrically actuated MEMS tunable lenses with various pupil geometries," Optical Engineering 56(3), 035104 (21 March 2017).
Article 3: Mahmoud A. Farghaly, Ulrik Hanke, Muhammad Nadeem Akram, Einar Halvorsen, "Trial functions for reduced-order models of piezoelectrically actuated microelectromechanical systems tunable lenses," Optical Engineering 57(9), 095103 (24. September 2018).
Article 4: submitted to Journal of Intelligent Material Systems and Structures Mahmoud A. Farghaly, Vladimir Kartashov, Muhammad Nadeem Akram, Einar Halvorsen, "Electromechanical modeling of piezoelectrically actuated MEMS tunable lenses with geomertic nonlinearity".