## Abstract

Considering the special characteristics of the removal function with the ring-shaped profile in fluid jet polishing (FJP), we present an effective method called the discrete convolution algorithm to compute the dwell function for controlling the figuring process. This method avoids the deconvolution operation, which usually fails to converge. Then an experimental confirmation of FJP figuring was demonstrated by machining a one-dimensional depth profile on a flat sample. The profile was figured from
$0.914\lambda \left(\lambda =632.8\text{\hspace{0.17em} nm}\right)$ peak to valley (PV) to
$0.260\lambda $. This experiment demonstrated the successful implementation of the algorithm to solve the dwell function in optical manufacturing.

© 2006 Optical Society of America

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### Equations (14)

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(1)
$$\mathrm{d}Z=kP{K}_{\nu \prime}\mathrm{d}t\mathrm{.}$$
(2)
$$\Delta {Z}_{\left(x\text{,}y\right)}=k\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{t}{P}_{\left(x\text{,}y\text{,}t\right)}{{K}_{{\nu}^{\prime}}}_{\left(x\text{,}y\text{,}t\right)}\mathrm{d}t}.$$
(3)
$$\Delta Z\left(x,y\right)=D\left(x,y\right)\text{\hspace{0.17em}}\ast \ast \text{\hspace{0.17em}}R\left(x,y\right).$$
(4)
$$\Delta {\tilde{Z}}_{\left(\xi \text{,}\eta \right)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\tilde{D}}_{\left(\xi \text{,}\eta \right)}{\tilde{R}}_{\left(\xi \text{,}\eta \right)},$$
(5)
$${D}_{\left(x\text{,}y\right)}={F}^{-1}\left(\frac{\Delta {\tilde{Z}}_{\left(\xi \text{,}\eta \right)}}{{\tilde{R}}_{\left(\xi \text{,}\eta \right)}}\right).$$
(6)
$${D}_{\left(x\text{,}y\right)}={R}_{\left(x\text{,}y\right)}{{B}_{\left(x\text{,}y\right)}}^{-1},$$
(7)
$${z}_{i}={h}_{i}{d}_{i}.$$
(8)
$${z}_{i}={h}_{i}{d}_{i}={\displaystyle \sum _{k=0}^{m}{h}_{k}{d}_{i-k}}\text{\hspace{0.17em} \hspace{0.17em}}\left(i=0,1,\dots \text{\hspace{0.17em}},m+n\right).$$
(9)
$$\left[\begin{array}{cccccc}{h}_{0}& 0& 0& \cdots & \cdots & 0\\ {h}_{1}& {h}_{0}& 0& 0& \cdots & 0\\ {h}_{2}& {h}_{1}& {h}_{0}& 0& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {h}_{m}& {h}_{\left(m-1\right)}& {h}_{\left(m-2\right)}& \cdots & {h}_{\left(m-n-1\right)}& {h}_{\left(m-n\right)}\\ 0& {h}_{m}& {h}_{\left(m-1\right)}& \cdots & {h}_{\left(m-n\right)}& {h}_{\left(m-n+1\right)}\\ 0& \cdots & {h}_{m}& \cdots & {h}_{\left(m-n+1\right)}& {h}_{\left(m-n+2\right)}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0& 0& \cdots & 0& 0& {h}_{m}\end{array}\right]\text{\hspace{0.17em}}\left[\begin{array}{c}{d}_{0}\\ {d}_{1}\\ {d}_{2}\\ \vdots \\ {d}_{n}\end{array}\right]\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left[\begin{array}{c}{z}_{0}\\ {z}_{1}\\ {z}_{2}\\ \vdots \\ {z}_{m}\\ {z}_{m+1}\\ {z}_{m+2}\\ \vdots \\ {z}_{m+n}\end{array}\right].$$
(10)
$${H}_{\left(n+m\text{,}n\right)}{D}_{\left(n\text{,}1\right)}={Z}_{\left(n+m\text{,}1\right)}.$$
(11)
$${D}_{i+1}={D}_{i}+kH\left({Z}_{i+1}-{Z}_{i}\right).$$
(12)
$${K}_{{\nu}^{\prime}}={\left(\frac{\nu}{{\nu}_{0}}\right)}^{2}=\left[1+{\left(\frac{\omega r}{{\nu}_{0}}\right)}^{2}\right].$$
(13)
$${K}_{{\nu}^{\prime}}={\left(\frac{\nu}{{\nu}_{0}}\right)}^{2}=\left[1+{\left(\frac{\omega r}{{\nu}_{0}}\right)}^{2}\right]=1+0.0759{r}^{2}.$$
(14)
$${d}_{i}\prime =\frac{{d}_{i}}{{K}_{{\nu}^{\prime}}}\text{\hspace{0.17em}}r.$$