Dwell time algorithm in ion beam figuring

Abstract

To get a high-precision optical surface, the deconvolved process of dwell time was transferred to a matrix equation in which the damped factor and the extra removal amount were introduced to expand the freedom of solution. A path weight factor and a surface error weight factor were used to take the scanning path and the initial surface error into account. Combined with the Gerchberg bandlimited extrapolation algorithm for initial surface error map extension, a high-precision final surface could be obtained within a factual aperture. Two surface error maps were calculated to rms = 0.1 nm from rms = 130.23 nm and to rms = 0.08 nm from rms = 82.74 nm. The simulations show that a perfect dwell time solution could be obtained by the revised matrix equation and initial surface error map extension with the help of the least squares QR (LSQR) algorithm.

Dwell-time algorithm for polishing large optics

“…For the dwell-time algorithms based on the linear equation method [18][19][20][21][22][23][24][25], although they can acquire surface form accuracy like this or even higher, the solution speed is too slow. For example, in reference [18], the solution of the dwell time is based on the following equation: r n r ×1 B n r ×n t t n t ×1 ;…”

“…Therefore, the simulation experiment is usually adopted to verify the dwell-time calculation algorithm instead of the actual one [22,23,25,32]. In references [22,23,25,32], all the initial surface errors are measured from a real part to make the simulation process closer to reality.…”

“…9(b). The final damping factor ξ 0.3235(0.3235 0.95 22 ), and the RMS of the residual error is 11.3 nm. It is noted that the convergence ratio still reaches 106.2 which is quite high.…”

“…To achieve a high-accuracy surface, calculation of the dwell-time algorithm plays a crucial role during the process. Hitherto, many algorithms to optimize the dwell time of CCOS have been studied [2,[12][13][14][15][16][17][18][19][20][21][22][23][24][25] and are mainly based on two models, which are the discrete convolution model and the linear equation model.…”

“…However, both of these methods still need a lot of resources to calculate which makes them not suitable for largescale calculations. Later, Wu et al [22] and Li et al [23] revised the linear matrix with a proper optimization operator in order to pursue faster convergence and edge protection, but the element number of the matrix combining the rate of material removal at all dwell points is extremely huge when encountering a large mirror, which means that its requirement for computer RAM would be rather high. Shi [24] chose an iterative method of a large sparse matrix to improve efficiency and accuracy.…”