## OHP Preprint No. 101 : Thin shallow meniscus mirrors

# Influence functions of a thin shallow meniscus-shaped mirror

### Luc F. A. Arnold

Laboratoire d'Astrophysique Observationnelle, Collège de France
and
Observatoire de Haute Provence (CNRS)
F-04870 Saint-Michel-l'Observatoire, France
**E-mail**: arnold@obs-hp.fr

Applied Optics (in press)

**Abstract**

Thin shallow spherical shell theory is used to derive the general influence
function, due to uniform and/or discrete (actuators) loads, for a thin
shallow meniscus-shaped mirror of uniform thickness, with a central hole and
supported at discrete points. Small elastic small deformations are
considered. No symmetry on the load distribution constrains the model.
Explicit analytical expressions of the set of equations are given to
calculate the influence functions. Results agree with Finite Element
Analysis (FEA) to within 1%. When FEA required megabytes of RAM memory, the
analytical method only needed kilobytes and ran typically 30 times faster.
This is a crucial advantage for iterative optimization of mirror supports
such as large passive or active meniscus-shaped primary mirror supports, or
Cassegrain/Gregorian adaptive secondary actuators configurations. References
are given that estimate the shear effects (thick mirror), the thickness
variation effect and the influence of the size of the support pads.
**Keywords** :
Telescope mirrors, mirror deformations, active optics, influence
functions, meniscus-shaped mirrors.

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