 
Summary: Uniformload and actuator influence
functions of a thin or thick annular mirror:
application to active mirror support optimization
Luc Arnold
Explicit analytical expressions are derived for the elastic deformation of a thin or thick mirror of uniform
thickness and with a central hole. Thinplate theory is used to derive the general influence function,
caused by uniform and@or discrete loads, for a mirror supported by discrete points. No symmetry
considerations of the locations of the points constrain the model. An estimate of the effect of the shear
forces is added to the previous pure bending model to take into account the effect of the mirror
thickness. Two particular cases of general influence are considered: the actuator influence function
and the uniformload 1equivalent to gravity in the case of a thin mirror2 influence function for a ring
support of k discrete points with kfold symmetry. The influence of the size of the support pads is
studied. Amethod for optimizing an active mirror cell is presented that couples the minimization of the
gravity influence function with the optimization of the combined actuator influence functions to fit
loworder aberrations. These lowspatialfrequency aberrations can be of elastic or optical origin. In
the latter case they are due, for example, to great residual polishing errors corresponding to the soft
polishing specifications relaxed for cost reductions. Results show that the correction range of the active
cell can thus be noticeably enlarged, compared with an active cell designed as a passive cell, i.e., by
minimizing only the deflection under gravitational loading. In the example treated here of the
European Southern Observatory's New Technology Telescope I show that the active correction range can
