In order to study correcting capabilities for figure errors of ultra-thin mirrors with active supports

a 0.5 m demonstration mirror is designed and tested. Firstly

the relevant relations between actuator forces and figure forms are analyzed

and some lower-order figure errors

such as primary spherical aberration

coma

astigmatism and distortion due to gravity

are introduced. Then

the optimization objective is set up

and the Sequential Quadratic Programming (SQP) method for nonlinear constraint problem is applied to calculating optimum actuator forces. The residual figure errors of the ultra-thin mirror are given. Simulation analysis shows that by using the actuator arrangement proposed in this paper

the normalized primary spherical aberration

coma

astigmatism as well as their summation can be corrected to be less than λ/24 rms

and the correcting capability for astigmatism is the strongest and for spherical aberration and coma are less; moreover

the deformation due to gravity combined with the summation of above three lower-order aberrations can also be corrected to be less than λ/24 rms. After the actual surface errors from the interferometry based on the 0.5 m demonstration mirror are given

the optimum actuator forces and form errors are solved. The results show that the surface quality is about λ/7 rms

which is in accord with the test data. The causes that the figure error can not meet the desired objective are analyzed

then an improved method is proposed. By the method

the figure error of the 0.5 m ultra-thin mirror with active supports satisfies the requirements equal to or less than λ/20 rms.