### Verification of characterizing MSF errors with the residual ripple

Spatial frequency of surface errors is divided into several separate bands in the field of high power lasers [2]: surface figure (>33 mm), MSF error (0.12 ~ 33 mm) and surface roughness (0.01 ~ 0.12 mm). There are two types of specification for MSF error; one is RMS value after band pass filtering, and the other is a not-to-exceed line for the power spectral density (PSD) as a function of spatial frequency [16]. In the following, we select RMS after band pass filtering over 0.12 ~ 33 mm range for evaluation of the MSF error.

MSF errors induced by CCOS processes are commonly in form of residual ripples. Thus, in order to quantitatively specify the correlation between residual ripple error and MSF error, we formulated a series of sinusoidal surface forms with variable spatial frequency and magnitude (see Fig. 1). The surface forms are sinusoidal distributed in *x* direction, while are uniformly distributed in *y* direction. Surface forms of this shape are fairly similar to the local regions of surface forms practically corrected by CCOS processes, which are nearly sinusoidally-distributed in the scanning direction while nearly uniformly-distributed in the feeding direction. Herein, the MSF error, in terms of the RMS value in the mid-spatial frequency band (0.0303 ~ 8.33 mm^{−1}), is calculated for all the surface forms as shown in Fig. 2. It is revealed that the spatial frequency has little effect on the RMS value, while there has a good linear relationship between the spatial magnitude and the RMS value. Thus, we should focus on the residual ripple magnitude rather than the frequency while restraining MSF errors.

### Influencing factors of residual ripple errors

As revealed above that residual ripple error can be characterized by the ripple amplitude, i.e. the peak-to-valley value of the ripple (PVe). We introduce a normalized PV value of residual ripple (PVn), which is derived from PVe divided by the average removal depth (r). PVn represents the residual error PVe while achieving unit removal (see Eq.1).

$$ {\mathrm{PV}}_{\mathrm{n}}={\mathrm{PV}}_{\mathrm{e}}/r $$

(1)

Primary factors impacting the residual ripple errors include the scanning pitch (i.e., path pitch), removal depth, and tool influence function (TIF) features. Without loss of generality, we modelled variable scanning pitch resulting in a uniform removal map as well as variable removal map under the same scanning pitch, to reveal the effects of the canning pitch and removal depth on the residual ripple and MSF error. Herein, we consider a Magnetorheological Finishing (MRF) TIF traversing a uniform pitch raster path, under the condition that the feeding direction is set in perpendicular to the fluid flow direction as shown in Figs. 3 and 4. As the TIF traverses a single line path with a constant feeding velocity of *v*, the removal is uniformly distributed along the feeding direction, while the removal distribution *R*
_{
j
} in the perpendicular direction can be obtained by Eq.2 (see Fig. 5), in which TIF matrix (*R,* unit in um/s) has *s* row, *k* column elements as shown in Eq.3; and the pixel size is *p* (unit in mm).

$$ {R}_j=p/v\cdot \sum_{i=1}^k{r}_{i,j},\kern0.5em j=1,\dots, l. $$

(2)

$$ R=\left[\begin{array}{l}{r}_{11}\kern0.75em {r}_{12}\cdots \kern0.5em {r}_{1k}\\ {}\\ {}\\ {}\\ {}{r}_{11}\kern0.75em {r}_{12}\cdots \kern0.5em {r}_{1k}\\ {}\\ {}\\ {}\\ {}\cdots \kern0.5em \cdots \kern0.5em {r}_{i\ j}\cdots \\ {}\\ {}\\ {}{r}_{s1}\kern0.75em {r}_{s2}\cdots \kern0.5em {r}_{sk}\end{array}\right] $$

(3)

Figure 6 shows the local removal amount distribution in the scanning direction while correcting uniformly-distributed form errors. The blue sections represent the removal amount in independent single path and the red one is the convolved removal amount. It is obvious that the convolved removal amount is periodically distributed, and the spatial wavelength is identical to the scanning pitch. It is confirmed that surface form correction by small-sized TIF inevitably induces residual ripple error.

Figure 7a shows that PVe becomes a linear growth along with the increment of the removal amount. It is suggested that a less removal amount is propitious to restraint of PVe. Figure 7b shows the PVn value as a function of scanning pitch. PVn increases as the scanning pitch is increased. It is noticeable that PVn increases slowly until the scanning pitch reaches ~1.1 mm, and then increases sharply. It is revealed that while correcting the surface form of optics by sub-aperture polishing, it is desired to adopt a smaller tool-path pitch for restraint of residual ripple.

### Development of the multi-pitch tool path

Correction of the form error by CCOS processes aims to polish every region to a desired plane of absolute flatness, which is commonly located at the lowest point on the surface as shown in Fig. 8. In fact, a lower plane has to be selected due to the maximum motion speed of the tool. Such a removal map introducing extra removal isn’t propitious to restraint of residual ripples as revealed above.

If the desired plane selected at the lowest point, the desired removal amount at the point would be zero. As the tool traverses across the point, it inevitably removes material deteriorating figure convergence, thus the tool are commonly driven with a most velocity allowable for the machine. Furthermore, the path pitch within lowest regions should be as large as possible so as to introduce less over-removal, but in uniform pitch tool path, a large pitch would deteriorate the residual ripple errors. Therefore, we develop a multi-pitch tool path which has a large pitch in less removal regions reducing over-removal and small pitch in more removal regions so as to decrease the residual ripples while guaranteeing the figure convergence.

The polishing procedure with the multi-pitch tool path is showed in Fig. 9. First, we should generate the removal map according to the actual surface figure and the desired surface figure. Then, the removal map is divided into several subregions based on the removal variance. After that we calculate the scanning path pitch and generate the path for each subregion. The spacing between adjacent dwell points along each path line, i.e. the feeding pitch, is also determined. The feeding pitch can be adopted within a wide range, yet value of the scanning pitch is recommended. After determination of the scanning and feeding pitches, we then acquire the dwell points on the whole surface. The polishing time at each dwell point (i.e. the dwell map) can be solved with various algorithms such as discrete convolution model, the linear equation model and so forth [17]. Finally, the CNC code can be generated according to the dwell point on the path and the dwell time map.

### Determination of the path pitch in each subregion

While generating multi-pitch tool path, we first divide the optic surface into several subregions according to the removal map. The whole material removal scope within the maximum and minimum removals is divided into several ranges, and then each removal range determines the corresponding subregions. The number of the removal ranges or subregions depends on the whole removal scope; the larger the removal, the more the ranges or subregions. Generally, 3 ~ 6 removal ranges or subregions are appropriate for most cases. Assuming a removal map has a maximum removal of *r* and a minimum removal of 0, it is divided into *m* subregions and the removal variance in each subregion has the same value d*r*, then the removal in each subregion can be derived by Eqs.4–5.

$$ \left(k-1\right)\cdot \varDelta r\le {r}_k<k\cdot \varDelta r,\kern1em k=1,\dots, m. $$

(4)

$$ \varDelta r=r/m $$

(5)

While determining the path pitch in a subregion, a dwell point *P* which has a removal of *h* and covers a square area in the subregion is considered, as shown in Fig. 10. The removal is almost uniformly distributed in the tiny square area, and then the correlation among the removal depth (*r*), path pitch (*d*) and feeding velocity (*v*) can be obtained by Eq.6. It is revealed that a certain *d* can be calculated for a given *s*, *r* and *v*
_{max}, as shown in Eq.7.

$$ s=r\cdot d\cdot v $$

(6)

$$ d=s/\left(r\cdot {v}_{\mathrm{max}}\right) $$

(7)

In Eqs.6-7 *h* represents the feeding pitch, *v*
_{max} the largest feeding velocity allowed by the machine, and *s* is the volume removal rate of the TIF, which can be derived by Eq.8:

$$ s={p}^2\cdot \sum_{j=1}^l\sum_{i=1}^k{R}_{i,j} $$

(8)

*Where p* (unit in mm) is the pixel size of the TIF, and Ri,j (unit in um) is the TIF removal rate.

As revealed in the previous section, minimum path pitches are desired for restraint of residual ripple errors. Eq.7 indicates that the feeding velocity is inversely proportional to the pitch; thus, we can adopt a maximum feeding velocity allowed by the machine so as to decrease the pitch. However, increasing feeding velocity has a significant impact on the stability of TIF. A too large feeding velocity will result in alteration of TIF, and hence deteriorate efficiency of figure correction as well as MSF errors. Further, the machine imposes restrictions on the moving velocity and acceleration of every movable component. Hence, there is a favorable maximum velocity allowed for each polishing machine. Herein, the largest feeding velocity (*v*
_{max}) allowed by the machine can be adopted in practice so as to reduce the pitch and hence the PVe.

As each subregion is determined within a material removal range, we adopt the minimum removal depth in each region for calculation of the corresponding path pitch (see Eq.9), which will prevent the feeding velocity exceeding the specified maximum value. Then, the pitch in each region can be obtained by Eq.10.

$$ {r}_1=0.5\cdot \mathrm{d}r,{r}_k=\left(k-1\right)\cdot \mathrm{d}r,\kern0.5em k=2,\dots, n. $$

(9)

$$ {d}_k=s/\left({r}_k\cdot {v}_{\mathrm{max}}\right) $$

(10)

In CCOS processes, the scanning path pitch should be restricted within a range in practice. The minimum value of the pitch is determined by the positioning & moving precision of the polishing machine. The maximum one is primarily dependent on the TIF size (i.e. <1/6 size). Further, a too large path pitch isn’t propitious to correcting the form error and restraining the ripple error.

### Solution and implementation of dwell time map

Dwell time map in terms of the polishing time at each dwell point provides the time that the tool dwells on the corresponding position to obtain desired removal. In the multi-pitch tool path, the dwell points are allocated at each path line with a feeding pitch. The feeding pitch can be specified according to the scanning pitch. Then, the dwell time map is solved by any developed algorithms, such as the discrete convolution method, linear equation method and so forth. The local feeding velocity (*v*
_{f}) can be derived from the pitch and the local removal (*r*
_{f}) at the corresponding point, as revealed in Eqs.11–12. In the multi-pitch tool path, the removal variance in each region is greatly decreased compared to the conventional tool path with a constant pitch on the whole optic surface. As the tool scans the path lines in any subregion, the path pitch is decreased as much as possible in every region, which is prone to improving the implementation precision of the dwell-time.

$$ s\cdot t={r}_f\cdot d\cdot {v}_f\cdot t $$

(11)

$$ {v}_f=s/\left(d\cdot {r}_f\right) $$

(12)

During generation of multi-pitch tool path, the optic surface is divided into several regions. In each region, the tool scans a raster path with a featured constant pitch. The pitch is dependent on the removal in the region, and the larger the removal, the smaller the pitch. During implementation of the dwell time map, the tool will traverse all the paths that generated covering the whole surface. Herein, we suggest that each region be scanned individually. In each region, adjacent path lines can be interconnected at the ends during implementation of the dwell-time map. As the tool traverses a path line and reaches the end, it translates to the nearby end of the next line and traverses this line (see Fig. 11). The translation stroke from one line to another maybe introduces extra dwell time, which will cause undesired removal and deteriorate the convergence rate of the surface form. If the tool lifts up after completing the last feeding segments in each path line, it will inevitably introduce extra removal during the lifting process. It is suggested that the tool lifts up while traversing the last feeding segment within a period longer than the determined dwell time. At this condition, the increased actual dwell time will compensate the decreased removal function achieving approximately the desired removal. Similarly, the tool descends while traversing the first feeding segment of the next path line. After the tool has covered one subregion, it also lifts off the optic and translates above to the first dwell point of another subregion. Then it descends to accomplish the subsequent dwell time. The lifting of the tool during the translation process wouldn’t bring with extra removal.