Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis

Abstract

An electromechanical model of a deformable mirror was developed as a design tool for adaptive optical systems. The model consisted of a continuous, mirrored face sheet driven with multilayered, electrostrictive actuators. A fully coupled constitutive law simulated the nonlinear, electromechanical behavior of the actuators, while finite element computations determined the mirror's mechanical stiffness observed by the array. Static analysis of the mirror/actuator system related different electrical inputs to the array with the deformation of the mirrored surface. The model also examined the nonlinear influence of internal stresses on the active array's electromechanical performance and quantified crosstalk between neighboring elements. The numerical predictions of the static model agreed well with experimental measurements made on an actual mirror system. The model was also used to simulate the systems level performance of a deformable mirror correcting a thermally bloomed laser beam. The nonlinear analysis determined the commanded actuator voltages required for the phase compensation, and the resulting wavefront error.

Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror

“…With the assumption 2h∕a ≪ 1, the normal strains S r and S θ of actuator can be expressed as −z∂ 2 u z ∕∂r 2 and −ðz∕rÞ∂u z ∕∂r, respectively. 15 Because the plate is thin, T 23 and T 31 are much smaller than the in-plane principal stress components and are thus neglected. 12,14 The axis-symmetrically deflection requires u θ ¼ 0 and ∂ð·Þ∕∂θ ¼ 0, which yields T 12 ¼ 0.…”

“…An FEA model of a CDPDM, which also has 21 elements and was modeled on a DM in Ref. 15, is shown in Fig. 7.…”

A temperature-dependent hysteresis model for relaxor ferroelectric compounds

“…For operating regimes in which thermal activation is negligible and relaxation times are small compared with drive frequencies, the local average polarization P is determined directly the minimization of the Gibbs energy (3). For the Helmholtz relation (1), enforcement of the necessary condition ∂G ∂P = 0 yields…”

“…A number of the initial models for the relaxor ferroelectric compound lead magnesium niobate (PMN), and derivatives doped with lead titanate (PT), barium titanate (BT) and strontium titanate (ST), focused on the quantification of the anhysteretic behavior above the freezing temperature T f -e.g., see [2,3,4,5,6]. Smith and Hom subsequently incorporated the hysteretic low-temperature behavior through the development of domain wall models having temperature-dependent coefficients [13,14].…”

Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines

“…Previous works modeling deformable mirrors have explored the physics behind the actuators and the mirror facesheet. Hom et al [5] presents a non-linear model of an electrostrictive DM, incurring a 13% underestimation error with respect to the real mirror position in a static test and ~40 nm rms errors when canceling a given wavefront. Andersen et al [6] and Laag et al [7] report results with a voice-coil actuator DM made by [8,9].…”