Electromechanical modelling of piezoelectrically actuated MEMS tunable lenses with geometric nonlinearity

Abstract

This article presents a variational model for the geometrically nonlinear behaviour of the piezoelectrically actuated MEMS tunable lenses. Residual stresses during fabrication and larger actuation voltages cause large deflections such that a linear model would provide less accurate approximation. This presses the need for a nonlinear model that can explain the softening and hardening effects exhibited by the lens during its operation and affect its optical performance. Thus, in the view of von Kármán’s plate theory, the presented nonlinear model predicts the lens displacement after solving a cubic nonlinear system of equations. The chosen displacement ansatz fits the problem under study by satisfying the mechanical boundary conditions, and simplifying calculation of the variational integrals and optical representation of the lens’ sag. The model also shows good agreement with FEM simulations over various combinations of tensile and compressive residual stresses. Moreover, it succeeds in fitting measurements when used in a constrained optimization scheme in which the layers’ residual stresses and the e-form piezoelectric coupling coefficient are the fitting parameters.