1. Introduction

The ever increasing demand for ultra-precise surfaces has motivated the development of varied deterministic polishing techniques. These techniques typically employ sub-aperture tools, moved along the surface to be polished in a controlled manner, in order to correct localized errors. For the process to be deterministic, the polishing parameters that control the tool influence function need to be precisely controlled. Computer controlled polishing, a technology that has been under development for decades [1], has reached maturity and allows for the precise control of each polishing parameter in order to converge towards high-quality surfaces [2]. There are currently several mature deterministic polishing techniques that can successfully produce meter-class, highly-precise surfaces. Magnetorheological finishing (MRF) [3, 4] uses magnetic fields to control the viscosity and thus, the removal rate, of a magnetic abrasive slurry. Ion beam figuring (IBF) [5] removal relies on energetic particle beams in a vaccum. The incidence of this beam onto the workpiece removes material in a controlled manner. ’Precessions’ [6] polishing removal is based on a spherical, inflatable rubber membrane covered with a flexible polishing surface, that conforms itself to the surface to be polished. In this polishing technique, the tool pressure and contact area can be controlled by varying the internal tool pressure and its axial position with respect to the part. A sub-aperture, rigid, conformal tool used to polish large surfaces has recently been developed [7]. This tool uses a non-linear visco-elastic medium that conforms to the surface to be polished.

All these techniques reliably control the polishing parameters that are important for that particular process to attain deterministic polishing. Most of these parameters remain stable for the time periods necessary to polish small optics, i.e. a few minutes. If, on the other hand, larger, meter class optics are to be polished, it is necessary to maintain the stability of these parameters during a polishing run, which can last several hours. This requires a very stable and precise control of the process.

In the case of the HyDRa polishing tool presented here, the process parameters slurry concentration, pressure and flux, as well as tool force have to be stably controlled to 1% for prolonged periods of time (over 100 hours), in order to guarantee a deterministic operation. We present a brief description of the HyDRa polishing tool and rig, the tool operational parameters, a discussion of why certain polishing parameters are key for the HyDRa system to polish deterministically and show finishing results of an 84 cm diameter mirror polished with HyDRa. Based on these results and taking other features of the HyDRa process into account, such as zero normal-force polishing, relatively high removal rates (13 mm³/h), small footprints that can minimize edge effects and correct small surface errors, low system cost and operation expenses, as well as the complete ability to conform to any surface shape (free-forms) [8], make HyDRa an interesting option in deterministic polishing.

2. The HyDRa polishing rig

HyDRa is an error-map based, deterministic, hydrodynamic polishing tool of the fluid jet polishing family (FJP) [9] that was developed at the University of Mexico [10]. Its internal geometry generates a tunable-density, high-velocity vortex that expels the slurry radially, grazing the work surface, and generating a footprint with central low-pressure and peripheral high-pressure zones that can be biased in such a manner that the tool exerts zero normal force onto the work surface. This tool is based on dwell-time and error-map correction and can achieve very high-quality surfaces. The tool mainly consists of three operational stages: a density control unit, one or more rotational acceleration stages and an output nozzle. The density control unit precisely controls the slurry to gas ratio, since slurry density is a determining factor of the abrasive particle velocity and consequent removal rate. The abrasive foam enters the second operational stage where it is rotationally accelerated and, when expelled through the throat, gains a high radial velocity due to the centrifugal effect. This generates an annular abrasion with tangential incidence onto the workpiece that tends to generate smooth, low-roughness surfaces. The principle of operation is discussed in more detail in [8].

The HyDRa tool is part of a complex polishing robot, consisting of slurry and air-supply subsystems, a slurry conditioning unit (SCU), a CNC-based tool positioning system, plus a series of interferometric surface-assessment instruments, as the block-diagram of Fig. 1 illustrates.

 

Fig. 1 The HyDRa system

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The HyDRa tool is attached to a five degree of freedom (DOF) polishing machine based on a 2.4 by 2.4 m Cartesian CNC, to which two additional DOF have been included by means of a 3-actuator hexapod that enables tip and tilt tool movements (Fig. 2). These five DOF allow generation and polishing of any surface geometry, including flats, spheres, aspheres and free-forms.

 

Fig. 2 The HyDRa polishing robot about to polish the 84 cm mirror. The upper left insert shows details of the HyDRa tool mounted onto the hexapod via a load cell.

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Since all polishing parameters are held constant by feedback-control, ideally, the removal effect is a function of dwell-time only. Thus, the figure correction is dependent on the trajectory followed by the tool and the velocity at each point along it. It is then necessary that the CNC be capable of following five-dimensional trajectories with controlled velocities while, in real-time, controlling z-axis movements to adapt to the surface with zero-force.

For this purpose, we entirely re-designed the original electronics, basing the new system on an industrial SBC with enough I-O ports to suit our present and future needs. The control program was written in C around a modified real-time LINUX kernel.

Although the machine repeatability is only ∼ 10 μm, the removal control has nanometric accuracy due to a load cell that regulates tool height over the workpiece, as reported in [8]. This allows the tool to follow the surface contour.

The slurry conditioning unit (SCU) is a system that supplies an adequate mixture of grit suspended in water to the HyDRa slurry-supply subsystem. It is also responsible of capturing and reincorporating the atomized and liquid slurry that is expelled through the HyDRa tool during the polishing process, maintaining it well stirred and with a constant density. The density is continually monitored with a photo-densitometer (see below) and controlled by means of polishing-paste and water supply systems. The return system consists of a 2 HP blower that suctions the expelled slurry and air and returns them to the SCU container, passing through a washed-air system, similar to a waterpipe. The blowing effect also aids in stirring the slurry. A pump is continually recirculating the polisher in the container and a derivation supplies filtered slurry to the HyDRa Slurry Control System (SCS). This system regulates the amount of slurry flowing into the tool by means of a damped diaphragm, DC pump that is controlled by feedback from a flux meter. In order to minimize air pollution, we polish in shallow immersion creating an air-lock for the return system.

As described in [8], the HyDRa tool accelerates a variable-density suspension of slurry and air (abrasive foam). This is achieved by means of an Air Control System (ACS) that regulates the foaming and propelling air pressures, with electro-mechanic air regulators, pressure sensors and ad hoc electronics. The SCS, ACS and SCU parameters are acquired with a data acquisition card, and visualized and controlled using LabView software.

3. HyDRa characteristic curves

For maximum polishing efficiency an abrasive foam is generated inside the tool. This allows to increase the velocity of the polishing particles, thus improving material removal. This foam is created by mixing slurry at a constant rate f of a few ml/s, and air, held at a constant pressure P_p. This mixture is then accelerated in one or more cylindrical chambers using pressurized air, at a propelling pressure P_T. The final mixture is expelled from the tool by a nozzle that produces a vortex which evolves into a radial flux, generating a uniform grazing footprint.

There is a relation of slurry flux (f) to slurry pressure (P_p) for each value of accelerating pressure (P_T). This further depends on the physical characteristics of the tool, such as general dimensions, type and shape of the accelerating chamber/s, as well as nozzle geometry. This relation constitutes an operational diagram that characterizes tool bias. One such diagram is shown in Fig. 3, where we can see that for each accelerating pressure P_T, the slurry flux f depends linearly on the supply pressure P_p. As can be seen from the figure, it is more convenient to control slurry flux instead of slurry pressure. We thus developed a slurry supply system based on a very precise flux-meter that is used as control feedback for the pump that feeds the slurry to the tool.

 

Fig. 3 Tool characteristic curves, where the relation between slurry flux (f) and injected slurry pressure (P_p) is shown for various accelerating pressures (P_T = 30, 40, 50 and 60 PSI). Linear fits are indicated in each case. The horizontal line represents operating points of HyDRa for different accelerating pressures P_T.

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4. Optical densitometer

One of the main parameters that has to be monitored and controlled during the polishing process is the concentration of polishing particles in the slurry mix. This concentration is traditionally measured by the relative density of the slurry to water, with the aid of a hydrometer, with the drawback that it cannot be used as electronic feedback. Since we could not find a suitable sensor in the market, we developed a low-cost optoelectronic device to perform this task in a continuous way. The device samples the polisher that is circulating in the polishing machine, on its way from the pump to the HyDRa polishing tool, giving a reliable record of slurry density. The device consists of an infrared LED light source and its corresponding photo-transistor detector that senses the optical transmission of the slurry as it passes through the densitometer. The signal decreases exponentially with the optical depth, as expected from the radiative transfer equation, F = F₀e^−τ; the optical depth given by τ = nσl, the product of the polishing particle density n, the cross sectional area σ of these particles, and the distance l between the source and the detector. This may be expressed in terms of the relative density of the polisher particles to water RD, which is related to the mass concentration of the grit, in g/l, and is ρ_i = ρ_H2O(RD − 1). From this, τ = K₁(RD − 1), where K₁ is a constant.

The current I at the photo-transistor depends linearly on the received flux, I = K₂F + I_D, where I_D represents the dark current and K₂ is a constant. It is thus expected that

The dark current I_D is of greater relative importance when the measured photo-current is small, as expected for the case of low illumination due to the large opacity of the slurry. In this case, variations of the dark current due to temperature fluctuations have to be accounted for. This is done by canceling this current with another signal from a matched blind photo-transistor at the input stage of the amplifier.

We measured photo-current as a function of the relative density RD. These measurements are shown in Fig. 4. As expected, the points are well fitted by a decreasing exponential curve. The values obtained for the fitting parameters are K₁ = 50, K₂ = 14 μA, and I_D = 0.2 μA. We performed a Kolmogorov-Smirnov (K-S) test to evaluate the relevance of the fit. From the data we obtain a K-S statistic of 0.43 that indicates a 0.99 significance level probability for the null hypothesis that the data is consistent with the model. This signal is used to regulate slurry density by feeding water and/or grit to the slurry management unit.

 

Fig. 4 Photo-current measured by the photo-transistor as a function of slurry concentration, given by its relative density to water RD. The solid line is a fit to an expected exponential decay in transmitted light intensity F.

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5. Deterministic polishing

Removal D of the HyDRa tool depends mainly on four independent operating parameters: grit mass concentration ρ_i, propelling air pressure P_T, slurry flux f, and height of the tool over the workpiece Z. In order to achieve deterministically polished surfaces, the errors contributed by each of these factors must be accounted for and these parameters need to be controlled accordingly.

The removal rates obtained during several independent experiments, where the polishing parameters were varied, are plotted in Fig. 5. In order to generalize the analysis, all parameters X are normalized around their operational values as X/X̄. Then, the relative variation of each parameter, defined as $\delta X=\frac{\mathrm{\Delta }X}{\overline{X}}=\mathrm{\Delta }\left(\frac{X}{\overline{X}}\right)$, is calculated. Δ is a measure of the variation of the parameter, which can either be a differential or the standard deviation. The differential is employed when a clear function is evident from the data. In this case the derivative is given by the quotient of the differentials, evaluated at the operational point and indicates the local slope of the function. However, if either the function appears highly non-linear or if the variation of the data is larger than the variation of the observed trend (cases d and c, respectively), then Δ is evaluated as the standard deviation of the data. This choice is consistent with adopting the worst-case scenario.

 

Fig. 5 Dimensionless normalized removal rate D/D̄ with respect to the normalized value of different polishing parameters (mass concentration ρ_i a), propelling pressure P_T b), slurry flux f c) and tool height z d)). The sensitivity of removal to each parameter is indicated in the upper left corner of each graph.

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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. Case a) of this figure shows the dependence of D/D̄ on normalized mass concentration ${\rho }_i/\overline{{\rho }_i}$. In this experiment we varied the relative density of the slurry (1 μm cerium oxide suspended in water) and measured the mass concentration ρ_i using the photo-densitometer calibrated as described above. In this case, 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 concentration.

The dependence of removal rate on the rest of the parameters is calculated in a similar way, where variations around the operation point of each parameter are taken into consideration. The sensitivities on propelling pressure P_T b), slurry flux f c) and tool height z d) are shown in Fig. 5. In this experiment, the operation points for these parameters were 40 PSI, 5 ml/s and 400 μm, respectively.

There is an optimal value of slurry flux for which removal rate is maximum and at which the HyDRa tools are operated. This value is around 5 ml/s and the removal only decreases slightly for higher or lower values, as can be seen in the example presented in panel c) of Fig. 5. For the case of tool height z, d), the force on the load cell F_c can be used alternatively, since, as shown in [8], tool force is an approximate linear function of height around the operation point, given that z = KF_c with K ∼10μm/Nw, and hence δz = δF_c.

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 $\delta D_T=\sqrt{{1.18}^2+2^2+{0.2}^2+{0.8}^2}/100=2.5\%$ or, in general

This implies that if 500 nm of material were to be removed in one polishing run, assuming parameters controlled to 1%, the total surface error would be 12.5 nm RMS, i.e. λ/50. This represents a 2.5% level of non-determinism (i.e. 97.5% determinism).

6. Polishing of an 84 cm mirror

Deterministic polishing of large surfaces requires for the removal rate to remain stable over prolonged periods of time. We accepted the challenge of polishing an 84-cm hyperbolic primary mirror in order to prove that HyDRa could deterministically finish meter-class optics [11]. This internally structured, lightweighted borosilicate mirror has a ROC of 4833 mm with a 1 cm thick faceplate. It had previously been polished using traditional methods to ∼ 2λ, peak to valley. In order to accurately measure the surface, a scanning Fizeau interferometer, able to measure optics of up to one meter in diameter, was developed at our lab [12]. The test rig was based on a classical configuration (Fig. 6) consisting of a transmission sphere, a null lens and a flat mirror used for autocollimation of the test surface. A Zemax simulation indicated that the main error contributions of this test rig to the wavefront were of the order of λ/50, assuming that the transmission sphere and the null lens are guaranteed to be ∼ λ/20 and the flat mirror ∼ λ/10. These values are expected for an on-axis symmetrical system. This interferometer allows us to measure surfaces with resolutions of 10 nm in the presence of structural vibrations, and up to 0.5 nm in stable conditions, by processing ∼ 100 images in ten scanning steps [12].

 

Fig. 6 Block digram of the metrology system used for measuring the 84 cm mirror.

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Surface measurements with this interferometer provided us with an initial 500 nm RMS error map. Apart from low-order aberrations, the previous classic polishing of the mirror induced a 50 nm print-through of the cell structure on the surface due to the thin face-plate, which was evident in our initial error maps. The mirror also initially presented large fallen edges of the order of 2μm.

Since HyDRa does not exert normal force on the surface, the mirror need not be supported in any particular way, but the metrology requires a stress-free whiffle-tree support, which was developed for this project. The HyDRa tool used for this experiment had a 7 mm diameter footprint with a removal rate on borosilicate of 13 mm³/hr. A two millimeter raster pattern, which was rotated by 4 degrees for each 10-hour polishing run, was chosen to minimize mid-spatial frequencies and polishing time. Polishing of this mirror consisted of three iterations of 5, 4 and 3, 10-hour machine-runs, respectively, for a total of 120 hours. The tool removal function was characterized for each iteration in order to keep track of tool wear (a typical tool useful lifetime is approximately 300 hours). The mirror was measured at the beginning of the polishing process to obtain the first error map, and after each iteration, so subsequent error maps could be obtained. Since this was the first time we polished a large surface with our system, the first iteration was intended to conservatively correct 50% of the initial surface errors. The two subsequent iterations were targeted to correct 80% of the remaining residual errors left from previous iterations. The last iteration could have been targeted to correct 100 %, but this could have resulted in an overcorrected surface; either way, the difference would have been within specifications. This conservative approach was to avoid over-corrections, force polishing runs to last ten hours and to be able to change tools in the case of excessive tool wear, which was not the case. Another reason was to be able to document the evolution of the polishing process.

In Fig. 7 the progress of the surface correction process is presented, with the first image a) representing the initial surface figure (500 nm RMS). From these images the dwell-time based trajectories were coded to generate the machining code for the next iteration. The next three panels show the results of the intermediate iterations with 305, 125, and 62 nm RMS; b) to d). The z-scale in all four cases is the same.

 

Fig. 7 Four error maps showing the progression of the polishing process. All maps have the same z-scale, shown in nm. a) shows the initial surface before polishing (500 nm RMS) and d) the final obtained surface (62 nm RMS). b), c) are intermediate iterations (305 and 125 nm RMS, respectively).

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As noted before, polishing with HyDRa can correct print-through left over from other, non zero-force polishing techniques. Since the HyDRa tool does not exert any normal force onto the surface, it naturally does not generate print-through due to any backstructure that may be present (Fig. 8), making it ideal for polishing thin mirrors, membranes or structured, lightweighted mirrors.

 

Fig. 8 The 84 cm mirror that was polished with HyDRa. a) shows a picture of the mirror showing the internal back-structure. b) shows the mirror prior to HyDRa finishing. Note the print-through left by the previous lap-polishing. c) shows the surface after HyDRa polishing. The print-through has entirely been removed by zero normal force error-map based polishing. In these figures, low-order Zernikes have been removed in order to highlight this effect. The mirror face-plate is 1 cm thick. Z-scales are the same and are shown as vetical bars in nm.

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The total removed volume was 960 mm³, amounting to a mean value of 1.6 μm removed over the entire surface. This however, was not enough to completely remove the original edge errors since, in order to achieve this, a much larger total volume removal would have been needed. Thus, the finishing process was terminated after reaching 62 nm RMS over the entire surface (λ/10 RMS, 0.7 λPV). Since our Linnik interferometer is not portable, we were unable to determine surface microroughness for this mirror, which was initially polished using conventional techniques to a specular surface. Qualitatively, the scatter did not worsen after HyDRa polishing. However, we consistently obtain typical microroughness values between 1 and 2 nm RMS on quartz using 1 μm opaline polisher, as reported in [8].

From these error maps, the level of determinism of the process could be evaluated. Each map allowed us to calculate a tool trajectory with specific dwell times. The total removed material was obtained by subtracting the previous error map from the measured one. Then, from the obtained result after polishing, we calculated the removed material for each iteration and plotted it as a function of dwell time. A linear relation is expected and the deviation from this represents the level of determinism.

In Fig. 9 we show this graph for the last iteration (3 runs= 30 hours). It shows a deviation of 10.6%, representing the obtained final level of non-determinism. The expected level of determinism of 97.5% could not be reached. We identified the problem to be due to non stability in the density control; one of the iterations presented fluctuations of up to 37%. The density parameter stability was difficult to control for extended time periods and did not reach the expected value of 1%, due to clogging and sedimentation issues, which have now been addressed. The rest of the parameters were controlled within 1%. 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 the mirror is progressively corrected, the surface is smoother and these changes in dwell time tend to decrease. This error source is not included in the tool level of determinism since it is not a tool parameter itself, but an error introduced by the CNC.

 

Fig. 9 Total material removed in the last iteration (3 runs= 30 hours) as a function of dwell time in each area element of 2.6×2.6 mm² (pixel size). The expected for a completely deterministic process is a linear relation at the removal rate of 13 mm³/hr (shown as a solid line). The actual deviations from this behavior amount to 10.6% which is the attained level of non-determinism and is the standard deviation of the points with respect to the best-fit line.

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This experiment was useful to evaluate the importance of the stability of each parameter in the level of determinism for prolonged time periods.

7. Conclusions

Polishing of this mirror was a success in the sense that we were able to prove that deterministic polishing over long periods of time is possible with HyDRa. We demonstrated that our low normal force technique easily and effectively eliminated the mount print-through artifacts. The mirror was polished up to its border, and we were able to correct most of the original edge problems. Due to time constraints, we could not go further correcting these edges.

As discussed previously, removal depends on the stable control of pressure, density, flux and tool height. In this experiment we were able to control three of four parameters to better than 1% during each run.

It has to be noted that the polishing rig used for this experiment was a prototype, so we sometimes had issues that forced us to stop the polishing run. When this occurred, a shallow groove was produced where the conditions were not the same after we resumed polishing, as is evident in some of the error maps of Fig. 7. These features became part of the error map in the next iteration and were mostly corrected. Our main problem was the stable control of the slurry density, causing fluctuations of up to 37 % in one of the iterations. According to equation 2, this would imply that in the last correcting iteration of the process, we had a contribution of 55 nm to the final error. We have taken several steps to correct this issue for future projects. In sum, we achieved a good surface (∼ λ/10 RMS) in a relatively good time.