## Abstract

The influence from the regular tool path to micro fabrication errors in deterministic finishing is studied through the simulations and experiments. The random pitch tool path based on the surface error distribution and the process parameters is designed to reduce this residual error when adopting the regular path to achieve the corrective polish. A nucleated glass flat mirror is polished with this method on the experimental installation UPF700-7 developed by ourselves. The surface accuracy is improved from the initial λ/30(RMS, 90%aperture, λ = 632.8nm) to the final λ/200 in 5 minutes and the medium-high spatial frequency errors induced by the regular path is restricted well at the same time. The accuracy of the simulation and the validity of the random pitch tool path are both proved through the experiments.

© 2010 OSA

## 1. Introduction

Magnetorheological Finishing (MRF^{®}) is a typical deterministic finishing technology commercialized by QED Technologies(R) for fine figuring of flat, spherical and aspheric optical components. It has the advantages of high process precision, good surface quality and high convergence efficiency [1–3]. The MRF^{®} fabrication principle is utilizing the rheological alteration of the MR fluid in the gradient magnetic field to create a flexible and extremely stable “sub-aperture polishing lap” that conforms to the optical surface. The work material is accurately removed and the surface error correction would be attained quickly by this lap based on the theory of computer controlled optics surfacing (CCOS) [4] firstly proposed by Itek inc. in 1970s. However, the micro fabrication errors called “fragmentary errors” are increasing in the process of fast surface error removal because that the “lap” size is less than the work piece like all of the other sub-aperture deterministic polishing technologies [5–7]. These medium-high frequency errors would be serious in some applications such as in intense laser systems and high resolution image formation systems which have the strict requirements for both of the surface error and medium-high frequency [8,9]. This error is mainly affected by the initial surface error distribution (spatial and frequency domain), the removal function characters (profile, removal efficiency and stability) and the adopted paths [10].

The experiences from the classical smoothing tell us this error would be restricted if the random tool path is adopted. The literature [10] also gives an explanation why the random tool path is favorable to reduce this kind of error in optics fabrication based on the entropy increase principle. The regular tool-path inevitably leaves respective signature in the surface, so the challenge is how to introduce elements of the randomness associated with the classical smoothing into the deterministic computer numerical controlled (CNC) process. Aiming at this problem, the partial random path which has added a random micro vibration on the vertical direction of the scan orientation is designed [10], but this method increases the implementation difficulty of the machine and sometimes it is impossible. The pseudo-random tool path is also presented for this problem and it is effective to reduce the error [11], but is hard to achieve a high precision on the surface error correction because of the difficulty on speed management. In this paper we first analyze the error sources when the regular tool-path is adopted in MRF^{®}, then a series of simulations and experiments are executed to validate the accuracy of the analysis. A process is done finally with the method of random pitch tool-path.

## 2. Error analysis

The removal depth in the optical surface at one location is the cumulated removal of polishing along the adjacent path lines in the deterministic finishing. The rationale can be approximately explained using the Eq. (1), *E* is the surface error, *r* is the removal function and *t* denotes as the dwell-time. The error will be removed completely if the convolution of the removal function and the dwell time equal to the surface error.

The surface error and the removal function data are both obtained through the interferometer while the dwell-time can be calculated with the various deconvolution algorithms in actual process. Then the CNC machine would control the “polishing lap” to remove the redundant material in the surface dwell point with different feed rate or residence time according to the dwell-time. Usually, this course will be treated for several iterations until the acquired accuracy is achieved finally. There will be no error theoretically if the “lap” is stable, the data is sufficient and the machine implements the correct dwell-time everywhere in the surface, but this cannot be real. The CNC machine will cross the whole surface with the process path and this path is usually a unicursal path. The raster and spiral tool paths shown in Fig. 1
are both the typical unicursal paths adopted in the deterministic finishing such as MRF process [12] to correct the surface errors. As shown in the figure, the scan speed *v*
_{1} can be continuous on these regular paths while the line feed speed *v*
_{2} is not. This is an important source for the medium-high frequency errors.

The surface accuracy can be high in existing fabrication results when the regular tool path is adopted, so we can research the relation between the process parameters and the residual errors to find out the optimal conditions to restrict it without changing these simple paths. The magnitude of the error from the tool path in MRF process is studied on the assumption that the removal function is completely stable and the errors from the measurement are neglectable. As referred previously, the residual ripple errors from the tool path are mainly contributed by the discontinuation of the line feed speed. We can simplify the problem just for thinking the errors on this direction. At the same time, the surface error is expressed using a series of discrete data where the dwell points stay, so we can analyze the problem with the simplified model shown in Fig. 2 .

Without loss of generalities, this model is established on the uniform material removal. The initial surface *S _{0}* is discretized as a series of adjacent data points

*p*(i = 1…n) and the distance between the dwell point equals to the pitch. The material removal

_{i}*E*(

*p*) on dwell point

*p*can be denoted as the Eq. (2). In this expression,

*r*(

*p*) is the material removal on point

_{i};p*p*when the removal function is staying on point

*p*, while

_{i}*t(p*indicates the dwell-time on point

_{i})*p*.

_{i}The material removal *E’(p’)* on non-dwell point *p’* is expressed with the Eq. (3). Then *r*(*p _{i};p’*) is the material removal on point

*p’*when the removal function is on point

*p*.

_{i}The dwell points value in polished surface *S _{1}* are the same when the removal function passes through the whole surface while the non-dwell points value are not. The residual error height $\sigma E({p}^{\text{'}})$ between them can be expressed with the Eq. (4).

## 3. Simulation and experiments analysis

#### 3.1 Simulation analysis

The residual error height $\sigma E$ is mainly associated with the removal function characteristics and magnitude of the dwell-time through the analysis in Eq. (4). We can find that the relationship between the value |*r*(*pi;p*) *-r*(*pi;p’*)| and the pitch is a direct proportion while the magnitude of dwell-time and the removal depth is also a linear relationship. Aiming at this circumstance, two actual removal functions shown in Fig. 3
and Fig. 4
are adopted to simulate the relationship between the residual error height and these influencing factors.

It is the removal function obtained with the polishing wheel whose diameter is 100mm shown in Fig. 3 and it is with the 200mm diameter wheel shown in Fig. 4. The left side in each figure is the 2D map and the right side is the profile along the north–south direction in the width direction center. Both of the removal functions immerse in the workpiece with the same depth 0.3mm and their peak removal value is also close. The geometry size of the first removal function is about 8mm × 6mm and the second one is about 15mm × 9mm.

The different material amount is removed utilizing the two “laps” with the same pitch. The simulation result in Fig. 5(a) shows that the residual ripple height becomes a linear growth along with the increment of the removal depth. Another simulation result shown in Fig. 5(b) is to adopt these two “laps” to remove the same material amount with the different pitch. The result shows that the residual error height grows quickly along the increment of the pitch. We can also find that the residual height with the first removal function is obviously larger than the second in spite of the same removal depth or the same pitch. This phenomenon can be explained with the previous analysis result, too. The volume removal efficiency with the first “lap” is clearly lower than the second one, so the more dwell time is needed on the one hand and at the same time, the adjacent dwell point value difference in the first “lap” is larger than the second one when the same pitch is used.

#### 3.2 Experiments analysis

A 100mm (90% effective aperture) diameter nucleated glass flat mirror was fabricated using the 1mm pitch along the raster path with the first removal function. The initial figure error and the polished figure are shown in Fig. 6
. The result shows that the figure error correction has achieved a notable convergence both on the peak to valley (PV) and root mean square (RMS) value. But we can see the obvious peaks at the spatial frequency 1mm^{−1} and 2mm^{−1} in the power spectral density (PSD) curve shown in Fig. 7
which is gained from the profile along the line feed direction shown in Fig. 6. The two-dimensional profile along the line from north to south in Fig. 6 is shown in Fig. 8
. The residual ripple height (PV) is about 7.5nm on area 1 and it is larger than value about 1.5nm on area 2. This is because that the material removal depth on area 1 is greater than the second position when the same pitch is adopted. All of these results show that the residual error because of the regular tool path is certainly existent and it is related with removal depth clearly.

Another mirror with the same maximum removal depth as the first mirror was polished adopting the 0.5mm pitch raster path. The removal function is also the same. The initial figure error and the polished surface map are shown in Fig. 9 and PSD curve of the line profile is shown in Fig. 10 . It shows the same rules as the previous experiment. The two-dimensional profile shown in Fig. 11 along the line from north to south in Fig. 8 indicates that the residual ripple height (PV) is about 0.9nm on area 1 and it is smaller than value about 2.2nm on area 2. These values are correspondingly smaller than the results shown in Fig. 8 where the same removal function and removal depth are applied. The results certificates that the smaller pitch is beneficial for limiting the residual errors caused by the regular tool path again.

#### 3.3 Strategy to reduce the ripple

The medium-high frequency errors caused by the regular tool path cannot be eliminated while could be restricted through the method of reducing the removal depth and pitch according to the simulation and experiment results. So the uniform removal and path pitch can be selected appropriately in response to the distribution of the surface error and process parameters in actual fabrication. The empirical formula that we choose is given as:

D means the removal depth, $\Delta P$ is the pitch value and $\sigma {E}_{r}$means the required residual ripple height. This formula is not convenient in practical application and we estimate with another process parameter${v}_{a}$ generally, which means the actual feed rate. It can be acquired through the specific management with the dwell-time and the identification is denoted as Eq. (6) if the raster path is adopted. The minimal feed speed ${v}_{\mathrm{min}}$ is based on the actual requirement and CNC machine performance.

It is favorable to change the pitch real-time according to the surface error distribution, so we can adopt the random pitch path in practical fabrication. The advantages for adopting the spiral path to manufacture the rotation symmetry optics are obvious than using the raster path according to the above analysis. The removal function sweeps the whole surface twice along the spiral path with the equilong pitch if the diameter-scan mode is applied. The actual pitch will be smaller than the spiral path pitch shown in Fig. 12(a) and the removal depth is just half of the whole depth every time. Thereby, it is a better choice to adopt the spiral path to polish the rotation symmetry optics. For example, the diagrammatic tool path we choose for the figure error correction is shown in Fig. 12(b). It is a typical pitch-variation tool path.

## 4. Experiment certification

Although the experience in optics fabrication indicates that the more unorderly the path introduce the smaller medium-high frequency errors, it is difficult to implement in MRF. Because the figure error correction convergence is the prime target, the tool-path must be appropriate to actualize the dwell-time. The random pitch path is verified hereinafter through the experiment. First, the nucleated glass flat mirror with 100mm diameter (90% effective aperture) was fabricated using the equilong 1mm pitch spiral path. The process time is about 30 minutes. The initial figure error and the polished surface are shown in Fig. 13
. It is shown that the figure error has been improved from the initial 0.334λ (RMS, λ = 632.8nm) to 0.059λ (RMS) after one iteration. But the periodic structure can be observed obviously on the polished surface map. The strongest peak is also observed on the 1mm^{−1} spatial frequency in the PSD curve shown in Fig. 14
. The result shows the same regularity as the raster scan.

Another fabrication on the same work was done with the random pitch spiral tool path along the diameter after the smoothing treatment and the result is shown in Fig. 15 . The whole process time is about 5 minutes. The figure precision has been improved to λ/200 (RMS). At the same time, there is no obvious ripple on the polished surface shown in Fig. 15 and it can also be found that there is no strong periodic feature in the PSD curve shown in Fig. 16 . The experiment has certificate the practicability of the random pitch tool path based on the surface error distribution and process parameters.

## 5. Conclusion

The rules from the process parameters to the residual errors when adopting the regular tool path are studied through the simulations and experiments. It certificates that the random pitch tool path based on the surface error distribution and parameters can restrict the tool path ripple height while the convergence efficiency would not be reduced.

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