HyDRa: Vortex Polishing with a Deterministic Hydrodynamic Radial Polishing Tool

2.1. The HyDRa polishing rig

The HyDRa tool is part of a complex polishing robot, consisting of a CNC positioning device; a fluid and compressed air control system; a slurry management unit, which stirs the slurry and controls its density; and a software package that obtains error maps from a series of interferometers, and generates dwell-time/constant-velocity PWM trajectories. These trajectories correct the workpiece figure, depending on the selected method (pulsed or continuous operation), as the block-diagram of Figure 2 illustrates.

The HyDRa tool is attached to a five degree of freedom (DOF) polishing machine with force feedback, based on a 2.4 × 2.4 m Cartesian CNC, with two additional DOF (tip and tilt), implemented by means of a 3-actuator hexapod. This configuration allows the generation and polishing of any surface geometry. Since feedback control keeps all polishing parameters constant, removal is exclusively a function of dwell-time. In this way, figure correction depends on the trajectory followed by the tool and the velocity at each point along it, which requires that the CNC be capable of following five-dimensional, controlled-velocity trajectories. Simultaneously, z-axis movement is controlled so that the tool can accurately follow the surface contour with zero-force. Although the machine’s repeatability is around 10 μm, the removal is accomplished with nanometric accuracy due to a load cell that regulates tool height over the workpiece, as reported in [4]. The slurry conditioning unit (SCU) supplies a density controlled polishing suspension to the HyDRa slurry supply system. It also captures and reincorporates the liquid and atomized slurry that is expelled from the tool. The density is continuously monitored by means of a photodensitometer and controlled by means of polishing-paste and water supply systems. The expelled slurry is captured by the return system, which consists of a blower that reincorporates this slurry and air into the SCU container, forcing the mixture through a washed-air system.

A pump is continually recirculating the polisher in the container, and a derivation supplies filtered slurry to the HyDRa slurry control system. The slurry that flows into the tool is fed by means of a damped-diaphragm DC pump that is feedback-controlled by means of either a flow meter or a pressure sensor. In order to reduce air contamination, the HyDRa system polishes in shallow immersion so that an air-lock for the return system is formed. The HyDRa tool accelerates an abrasive foam, composed of a variable-density suspension of slurry and air created by means of an air control system that regulates the foaming and propelling air pressures using electromechanic air regulators, pressure sensors and control electronics. All polishing parameters are acquired with a data acquisition card, and controlled and visualized in LabView®. The surface is fixed onto the CNC platform as the tool is swept over the surface in a pattern chosen by the user. Pulsed operation of the tool sweeps the tool(s) along the workpiece at a constant velocity, switching the polishing action on and off depending on how much material needs to be removed. The slurry is fed to the tool at controlled pressures and flow rates. The tool is supplied with compressed air to operate the foaming and acceleration stages. The abrasive foam is radially expelled through the tool nozzle onto the surface to be polished, as described above. Tool height is controlled by the feedback variable provided by the load cell, so that the CNC can adjust tool height in order to polish with zero force, as is discussed below.

2.2. Zero-force polishing

One of the central advantages of the HyDRa tool resides in that it can be adjusted to exert zero force on the work surface, while maintaining considerable removal rates (~10 mm³/h). The flotation effect is described in more depth in this section.

As the HyDRa tool expels the slurry through its nozzle, the rotational energy of the flow is converted to radial velocity. The drag generated by this radial flow produces a central low-pressure zone that is surrounded by a vortex which is in turn confined by a ring shaped, positive thrust zone. This effect differs from the linear jet polishing technique [3, 8] in that, in normal-incidence, classic jets, the pressure profile is represented by a Gaussian distribution, where the maximum is located at the jet’s center. In contrast, the pressure distribution of HyDRa on the workpiece presents negative values at the footprint’s center and is circumscribed by an annular, positive-pressure region. With this, the force on the workpiece can be adjusted to net zero-force. In Figure 3a, footprint pressure vs. distance from the center of the tool is plotted. This was obtained by means of a 0.46 mm diameter orifice connected to a pressure sensor over which the tool footprint was radially scanned. As can be seen, a low pressure zone forms at the central part of the footprint and, as the radius increases, the gauge pressure increases to a maximum. It then falls again as the orifice approaches the tool’s outer radius.

As the tool moves away from the workpiece, the vacuum has a tendency to increase, as the thrust force tends to decrease; in consequence, the balance of these two forces is a function of the tool’s distance from the surface to be polished. With this, several operational modes can be achieved, one of them allowing the tool to freely float over the surface without the need to rigidly attach it to a positioning device. In this case, the tool’s weight is counterbalanced by the thrust and vacuum forces. When the HyDRa tool is mounted onto a passive hexapod through a load cell, the support force Fh counterbalances the tool weight mg and the remaining forces to zero, while maintaining the tool in a static position.

The load cell value Fh is then used as feedback for the control system to maintain zero force on the workpiece. This is done by adjusting the z-axis of the CNC/hexapod.

We measured the tool and workpiece force interactions as a function of separation. In Figure 3b, the overall behavior of this interaction is shown in experiments where slurry accelerating pressures of 20 to 50 PSI and a polishing slurry flow of 5 ml/s were used. The distance over the workpiece was modified from several millimeters, down to a few hundred microns. The values of the load cell F_h were used to obtain the substrate force F_x, as explained above.

When the tool exceeds a certain distance, the workpiece experiences no force (see Figure 3). As the tool approaches the substrate, a negative force (attraction) is produced by a vacuum that develops between the workpiece and the HyDRa tool. As the tool further approaches the workpiece, a positive repulsion force is experienced, caused by the thrust force. At a few hundred μm from the workpiece, the vacuum and thrust forces balance each other out to zero. Here the surface can be polished without being deformed by the tool.

The combination of attractive and repulsive forces commonly arises in physical problems. One of such is given by the empirical Morse potential of diatomic molecules Uz=B1−e−αz−zm2−B, where B is the dissociation potential and zm is the separation at which the minimum of the potential is reached, which constitutes a stable equilibrium separation. A force can be readily obtained from this potential by calculating its gradient: Fz=−ddzUz. We have plotted this force along the data of the figure by least-squares fitting the three parameters, B, α and zm; their resulting values are listed in Table 1. A reasonable fit is obtained, and some insights may be derived from this exercise. It can be noted that, as the accelerating pressure PT increases, so does the depth of the potential well and thus the amplitude of the forces involved, as given by the parameter B. However, the minimum of the potential is found roughly at a constant separation z_m of around 340 μm. At this distance, the force derived from the potential is zero (since the derivative of the potential is zero) and the tool floats over the work surface in stable equilibrium. If we approximate the minimum of the potential by a square law (Hooke’s law) a restitution spring constant k = 2αB can be obtained, which measures the force with which the stable position is restituted. The values of k are also given in Table 1. As indicated in the inset in Figure 3b, k is a linear function of the accelerating pressure P_T.

2.3. Tool influence function and removal rates

Several tools have been developed to accommodate for different removal rates that range from 1 to 600 mm³/h. This section presents measurements of static and volumetric removal rates for a medium removal-rate tool with a nozzle diameter of 3.8 mm. Several polisher grits have been tested: Cerium oxide (Opaline) with a particle size of 1 μm, and aluminum oxide (μ-grit) with particle sizes of 5 and 12 μm, suspended in water at a constant relative density of 1.09. Samples of different materials were polished: standard window glass, water-free fused silica (Infrasil®-302), BK-7 borosilicate, and Ohara’s CLEARCERAM-Z® vitroceramic.

The measurements were performed by scan-sliding the tool over the sample at a constant speed and a corresponding zero-force tool height. The resulting cavity was measured interferometrically, and both the volume of removed material and the depth of the cavity were calculated. With these measurements the static removal rate DS (depth/time) and the volumetric removal rate DV were then calculated. We performed these measurements at different operational conditions of accelerating pressure (PT) and slurry flow (fp). In all cases, the relative density of the slurry was kept at 1.09. In Figure 4, the test results are shown. In all three panels, the left vertical axis DV is related to DS as DV = DS* A, where A is the tool footprint area. Each plot indicates the dependence of removal on the accelerating pressure PT. The top left panel, where Cerium oxide was used, slurry flow fp values of 3, 5 and 7 ml/s on window glass were tested. These are indicated as numbers on the curve.

As can be noted, the highest removal rate for this tool is achieved with a slurry flow of 5 ml/s, thus this flow was employed for all consecutive experiments. Removal rates up to 2.5 mm³/h at 50 PSI are obtained for this polisher. The fp = 5 points are well fitted by a second order polynomial. In the first panel we also show the results of polishing a fused silica sample (crosses). In this case the hardness of the material causes the removal rate to decrease to 66% compared to window glass. The dotted line that fits these values is obtained by multiplying the same DS polynomial by a small factor. In the lower left panel we tested the removal rate of 5 μm aluminum oxide grit on glass (crosses). The volumetric removal rate rises to ~50 mm³/h at an accelerating pressure of 50 PSI. The solid line is the DS polynomial obtained previously, multiplied by 19. Again, a very good fit can be seen. Lastly, in the lower-right panel, the removal rate of aluminum oxide grit (12 μm particle size) on Ohara CLEARCERAM® vitroceramic is shown. Removal was so large that it was impossible to measure it reliably with an interferometer, so the resulting cavity was measured by means of a needle profilometer. Only two points were taken in this experiment (crosses in the last graph). If the same polynomial DS were to be fitted (dashed line), it would have to be multiplied by 94. For all cases, the removed volume per abrasive particle is 3 to 5 orders of magnitude lower than the particle’s volume. The kinetic energy as well as the grazing incidence associated with both the cerium and alumina particles, creates stresses that are not large enough to produce permanent dents. Resulting typical micro-roughnesses of 2, 24, and 31 nm, for grit sizes of 1, 5, and 12 μm, respectively, are around three orders of magnitude smaller than the particle sizes. These tests demonstrate that independently of grit size and material, as well as substrate hardness, behavior of removal as a function of accelerating tool pressure is comparable, which indicates that the same material removal process discussed above is taking place. The scaling parameter is an indicator of how efficiently a particular abrasive removes material from a surface of a given hardness. All HyDRa tools present this same relationship of removal as a function of PT and grit size for pressures of up to 90 PSI. Currently, typical removal rates of ~15 mm³/h @ 90 PSI, using 1 μm Opaline on borosilicate glass, are common.

2.4. Software

The HyDRa trajectory planning tool (HyTPT) is a software package developed specifically for the HyDRa tool [9]. It feeds machining code to a CNC or any computer-controlled positioning device, based on the error map that has been obtained interferometrically. When only one HyDRa tool is available, the amount of material that is removed at each specific position is proportional to the dwell time. This dwell time can be controlled either by the speed of the tool along a given trajectory when the tool is operated in continuous mode, or by the width of the pulses when operated in pulsed mode.

In HyTPT (Figure 5) a main window presents the project name and grants the user the ability to go from error maps to machine coordinates in four steps:

1. Error Map alignment: This allows the easy alignment of the error map, and the determination of its center, orientation and pixel size. The basic shapes are rectangular, circular, and annular surfaces.

2. Base trajector: It is possible to select the base trajectory from a series of curves: from simple raster patterns to more complex curves, such as trochoids and rotating triangles. It is possible to define and analyze each curve from the point of view of trajectory density, total polishing time, and speed at each point, among several others.

3. Surface shape: The shape of the surface, currently any on- or off-axis conic section, is defined in this window. The 2-D base trajectory is projected onto this 3-D surface. In addition to the 3-D trajectory the normal vector to the surface at each position is calculated, so that the CNC may orient the HyDRa tool normal to the surface along the trajectory.

4. Machine coordinates: A set of alignment tools facilitates the transformation of surface coordinates to machine coordinates. These routines communicate with the CNC machine and are developed for each particular CNC or robot arm to which the HyDRa tool is attached. Finally, the centering and orientation parameters, nine in total, are determined in order to properly align the system. Once each of these parameters has been calculated, the trajectory is delivered to the CNC machine, or robot arm so it can be executed there.

3.1. Linearity

A pulsed HyDRa tool with a 7 mm footprint has been developed and tested for linearity. Figure 6a shows the results of erosion vs. dwell-time. Pulse width was varied at constant increments, starting from 10 ms to a maximum of 500 ms, as the tool was moved at 0.2 mm increments, overlapping 35 times at each tool footprint diameter. The erosion was measured using a Fizeau interferometer and the result was normalized, so that removal corresponded to a single pass of the tool over each point along the line that was polished. Error bars are primarily due to errors produced by the subtraction of the base reference during interferogram reduction. A removal resolution of 0.1 nm/ms can be seen from the data. Noticeable polishing effects were observed at 25 ms. We attribute this effect to the electro-valve response time, which can be improved by using faster actuators.

3.2. Pulsing the polishing process

3.2.4. Tessellated polishing

In the polishing of meter-class surfaces, efficiency is limited due to HyDRa’s small footprint size and volumetric removal rate. Efficiency, however, can be improved by simultaneously polishing the surface with several HyDRa tools. These tools can be mounted on independent polishing robots, where each robot tackles a certain section of the surface. Alternatively, several tools can be mounted on a single robot arm, as described below. This method poses several problems, such as obtaining smooth seams between sections, approaching each section without leaving marks, and avoiding collisions as two tools concurrently approach the boundary between sections. In order to obtain a seamless interface between two independent sections, it is necessary to approach the boundary following special trajectories, such as the wedge pattern shown in Figure 7 (bottom). This trajectory avoids duplicating dwell time at the seam, such as would happen if a rectangular pattern were used (top of Figure 7).

3.3. Multiple-head polishing

Since HyDRa tools can now be pulsed, several polishing heads can be mounted onto a common arm which moves at a constant velocity over the surface. Dwell time is then controlled using PWM for each tool, as required by the error map. We can also take advantage of the self-conforming capabilities of HyDRa, in that it is not necessary to employ a positioning device to conform the parallelism of the tool to the surface. Only one degree of freedom (DOF) per tool is required. Each loop is closed with a load cell signal and implemented by means of a linear stage, which permits zero-force polishing while freely following the local sag and tilt of the surface. Another advantage of this type of polishing is that one single slurry supply system can be used for all the tools, simplifying the system and considerably reducing the costs. Polishing efficiency becomes a function of the number of tools, and in the case of a matrix configuration, several polishing runs can be implemented into a single sweep, reducing polishing time.

By polishing with several tools, each tool is essentially given a section of the surface and the boundaries between sections are finished seamlessly, either by employing wedged joints, or by using PWM. Among the possible multi-tool configurations are matrix, linear and spiral layouts:

3.3.3. Spiral

In the case of large circular mirrors, it may be more efficient to polish the mirror by placing it on a rotary table. If we seek to fix as many HyDRa tools as possible onto a single arm, two conditions must be met. First, the number of tools should increase as r², so that each tool covers the same mirror area. Secondly, tools must be packed at the maximum allowable density, so that each tool sits next to the following one. Therefore, the shape of the arm must be a kind of spiral which is possible to solve for. By calculus of variations a spiral curve parameterized by the (increasing) radial coordinate, given in polar coordinates (r, θ) results in: θr=4k2−1/r2−arctan4kr2−1,r>1/2k.An example of this is shown in Figure 8, where HyDRa tools are mounted at equal separations along a spiral arm and may move along it to span equal areas, as the surface under it rotates.

4.1. Deterministic polishing

To maximize polishing performance, an abrasive foam is created in the tool’s first stage. This raises the velocity of the polishing particles, improving the removal of material. This foam is produced by combining a constant flow f (a few ml/s) slurry, with air that is kept at a constant pressure Pp. This fluid is then accelerated with pressurized air at a propelling pressure PT, in one or more cylindrical cavities. The resulting abrasive foam is then expelled through the tool’s nozzle, where a vortex is produced that develops into a radial flow, and generates a grazing, uniform removal footprint. A relation of slurry flow f to slurry pressure Pp exists for each value of accelerating pressure PT. This, in addition, depends on the tool’s physical characteristics, such as its overall dimensions, the geometry of the acceleration chamber(s), as well as the nozzle shape. This relation establishes an operational diagram that defines tool bias. In this section, the control of f is chosen, although it is possible to select to control for either f or Pp. The removal D of HyDRa mainly depends on four independent operating parameters: propelling air pressure PT, grit mass concentration ρi, slurry flow f, and distance of the tool over the workpiece Z. In order for deterministically polished surfaces to be obtained, the errors contributed by each of these parameters must be taken into account and controlled.

The removal rates, as determined by a series of independent experiments, where the polishing parameters varied, are shown in Figure 9.

To generalize the analysis, all parameters X are normalized around their operational values as x/x¯. With this, the relative variation of each parameter, defined as δX=ΔXX¯=ΔXX¯, is computed. The ratio between the relative variations of removal rate and the relative variations of each polishing parameter is given in the upper-left corner of the graphs. Here the operational value for the relative density is 80 g/l. As can be seen, removal rate varies as 1.18 times the fluctuations in the concentration. The relationship between the removal rate and the rest of the parameters can be determined similarly, where the fluctuations around the operation point of each parameter are taken into account. The sensitivities on propelling pressure PT (upper-right), slurry flow f (lower-left) and tool height z (lower-right) are 2, 0.3 and 0.8, respectively. In this test, the operation points were 40 PSI, 5 ml/s and 400 𝜇m, respectively. An optimum value of slurry flow (around 5 ml/s), for which the removal rate is maximized and which is chosen to operate the HyDRa tools, can be found. Removal only decreases slightly for higher or lower flow rates around this value. This is shown in the lower left panel of Figure 9.

In the case of tool height z, load cell force Fc can be used instead, since, as shown in [4] tool force is an approximate linear function of distance when close to the operation point, given that z + K Fc, with K~10 𝜇m/N, and hence δz = δFc..

If we assume that each of these four variables is statistically independent, the total error can be added in quadrature. For example, if each parameter is controlled to ~1% precision, then the total error δD_T is δDT=1.18∗δρ2+2∗δPT2+0.3∗δf2+0.8∗δz2, which amounts to 2.5% for this case. This means that if 500 nm of material are removed in one polishing run, considering that the parameters are controlled to 1%, the total surface error results in 12.5 nm RMS (λ/50). This constitutes a 2.5% level of nondeterminism (i.e., 97.5% determinism).

4.2. Polishing example: polishing of an 84 cm mirror

As mentioned before, in order to deterministically polish large surfaces it is imperative that removal rate remains stable over extended time periods. We polished an 84-cm hyperbolic primary mirror to λ/10 RMS, 0.7 λ PV in order to prove that HyDRa could deterministically tackle meter-class optics. The polishing process is described in [12]. From the error maps that were acquired during the iterations, the level of determinism of the process could be calculated. From each map we computed a tool trajectory with distinct dwell times. The amount of removed material was calculated by subtracting the previous error map from the measured one. Then, from the obtained result after polishing, the removed material for each iteration was determined and plotted as a function of dwell time. Refer to Figure 10. A linear relation is expected and the deviation from this represents the level of determinism, Figure 11. This experiment was useful to evaluate the importance of the stability of each parameter in the level of determinism for prolonged time periods. In the figure, a larger error can be noticed for shorter dwell-times than for longer ones. This is due to CNC errors when the tool has to be quickly accelerated to obtain short dwell-times. As the mirror is progressively corrected, the surface is smoother and these changes tend to decrease.

4.3. Polishing example: PSD and polishing of Fabry-Perot etalons

Three components in the power spectral density (PSD) of the residual surface errors that are related to the footprint diameter of the tool DH exist for any given polishing method. In the low frequency domain (L >> DH), the surface errors (optical figure) are a function of the stability of the polishing parameters during the polishing run, while at the high-frequency domain (L << DH), the physics of the polishing process determine surface quality (micro-roughness). In the case of mid-spatial frequencies (L~DH) surface quality depends on the geometry and overlap of the polishing trajectories. We obtained PSD measurements as described in [4] and references therein. The PSD is discussed using the results obtained while polishing four 50 mm etalon plates to better than λ/100. These 50 mm diameter water-free fused silica plates are used in an NIR scanning Fabry-Perot interferometer. These surfaces were polished using the HyDRa tool discussed in Section 4. Surface measurements were taken with a phase-shifting (PS) Fizeau interferometer in order to quantify the figure with a 180 μm pixel size projected onto the surface. A PS Linnik interferometer with 2× and 50× objectives (equivalent pixel sizes of 7.6 and 0.16 μm) was used to determine mid- and high-spatial frequencies, respectively. In Figure 11a, the 2-dimensional power spectrum PSD₂ vs. spatial frequency is plotted. It can be noted that three overlapping regimes exist that correspond to the series of instruments that were used to evaluate the surface quality. The integrated RMS values for each regime are 3.8, 1.5 and 2.9 nm for low-, mid- and high-spatial frequencies, respectively. The overall slope is approximately described as f−2.5. Since in this section the physics of the HyDRa tool are described, emphasis will be made on the PSD high spatial-frequencies (micro-roughness). For lower frequencies, i.e. meter-class optics, where it is crucial to achieve a very high stability of the operational parameters, refer to [13]. Figure 12b shows one of four etalon plates that were polished.

Interferograms [10] showed initial figure errors that ranged between 27 and 83 nm. Using these measurements, we calculated the error maps to compute a dwell-time based raster pattern trajectory for the CNC polishing machine. An acceleration pressure of 40 PSI was chosen and tool height was controlled to achieve zero-force on the workpiece. RMS surface qualities between 3.6 and 6.8 nm were obtained after two 15 min polishing runs. The low frequency interval of the PSD shows an overall RMS fit to the desired figure of 3.8 nm, which is in accordance with the results presented in the previous section: a final surface figure quality of >>λ/100 for visible wavelengths. Sub-aperture polishing can introduce unwanted patterns associated with the polishing trajectories [14] which can occur in HyDRa polishing with a 7 mm footprint on a 40 mm sample. To minimize these mid-spatial frequencies, the tool was raster-scanned with 0.25 mm steps, which corresponds to 1/20th of the tool’s footprint size. Traces of this raster pattern should have been visible in the 0.14–4 mm⁻¹ frequency range on the PSD. However, no evident peaks that could have been related to grooves left by a raster pattern could be observed, and only a tiny peak corresponding to about 100 μm could be noted. This peak adds a very small fraction of a nm to the total 1.5 nm RMS of the mid-frequency band, demonstrating that “over-rastering” can be effective in minimizing mid-spatial frequencies. This represents an alternative to the approach proposed by Fähnle [15], using only one tool. A comparatively large footprint has the extra advantage of making our process insensitive to CNC positioning errors, which are two orders of magnitude smaller than the footprint size. Finally, the high-spatial frequency domain shows an RMS of close to 3 nm. Although the PSD decreases in this region, it apparently stabilizes at frequencies >10³ mm⁻¹ or sizes smaller than 1 μm. This number is related to the grit size that was used in this test. To sustain a decreasing PSD tendency and thus, smaller values of the high-frequency RMS, the use of smaller grit sizes is suggested. Integrating the high-frequency domain of the PSD to obtain the RMS is also equivalent to calculating the RMS directly from a 50 × 50 μm area of the micro-interferogram, according to a standard definition of micro-roughness [16]. This micro-roughness obtained with the HyDRa process (3 nm) is comparable to the roughness reported in current FJP literature [5].