Modeling near-null testing method of a freeform surface with a deformable mirror compensator

We present and analyze a new measurement method of freeform surfaces with a deformable mirror (DM), which functions as a wavefront compensator. We discuss the maximum slope compensation of a DM in the first-order approximation and derive how to calculate the ideal DM form. A constrained optimization procedure is conducted to ensure the actual DM form fits the ideal DM form as closely as possible. To demonstrate the feasibility and understand the measurable range of our method, we evaluate freeform surfaces with a single Zernike term, the first 36 random Zernike terms, and 230 random Zernike terms. A tolerance analysis of the DM position is carried out to show the effect of the tilt and the decenter of the DM on the measurement accuracy. Our studies show that the proposed system is very flexible for freeform surface metrology.