Zernike polynomial based Rayleigh-Ritz model of a piezoelectric unimorph deformable mirror
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Zernike polynomial based Rayleigh-Ritz model of a piezoelectric unimorph deformable mirror Craig S. Long • Philip W. Loveday Andrew Forbes
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Received: 18 November 2011 / Accepted: 15 March 2012 / Published online: 3 April 2012 Ó Springer Science+Business Media, B.V. 2012
Abstract Piezoelectric bimorph- or unimorph-type deformable mirrors are commonly used in adaptive optics to correct for time-dependent phase aberrations. In the optics community, the surface deformations that deformable mirrors are required to achieve, are routinely and conveniently described using Zernike polynomials. A Rayleigh-Ritz structural model, which uses Zernike polynomials directly to describe the displacements, is proposed in this paper. The proposed formulation produces a numerically inexpensive model that predicts deformations with remarkable accuracy. Since design variables, such as electrode layout, material properties, and mirror dimensions, are represented analytically, the model is well suited to optimization or sensitivity analysis applications. Furthermore, since the numerical implementation is very efficient, it could be employed in closed-loop control C. S. Long (&) P. W. Loveday CSIR Material Science & Manufacturing, Box 395, Pretoria 0001, South Africa e-mail: [email protected] P. W. Loveday e-mail: [email protected] A. Forbes CSIR National Laser Centre, Box 395, Pretoria 0001, South Africa e-mail: [email protected] A. Forbes School of Physics, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
applications. Results achieved with the proposed model compare well with results from a traditional finite element analysis as well as experimental results of a representative design. Keywords Deformable mirror Piezoelectric unimorph Rayleigh-Ritz Zernike polynomial
1 Introduction Adaptive optical systems are used in large earth based telescopes to correct for the effects of atmospheric turbulence (Beckers 1993), in retinal imaging systems (Liang et al. 1997) and various other laser applications, for example (Vdovin and Kiyko 2001; Kudryashov and Samarkin 1995; Litvin et al. 2008; Forbes et al. 2008). The incoming wavefront is measured, usually using a Shack-Hartmann wavefront sensor for laser applications, and an aberration identified. A wavefront corrector in the light path is then used to remove or alleviate the phase error. Although various wavefront correctors are available, such as spatial light modulators or deformable lenses, the most popular wavefront correctors for accurate, low order correction, are deformable mirrors. Piezoelectric bimorph and unimorph mirrors, such as that schematically depicted in Fig. 1, are suitable for laser applications (Kudryashov and Samarkin
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C. S. Long et al. Reflective surface Individual electrodes
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Piezoelectric disc Copper disc with cavity in non−reflective side
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Fig. 1 Side and back view of the proposed unimorph mirror design
1995; Ellis 1999). Although they are usually more expensive and better suited for low-order correcti
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