Polished surfaces are usually used for corrective figuring in ultraprecision machining, but in this research, corrective figuring of a rough quartz sample has been preformed before the polishing process. A capacitor probe has been used to measure out the flatness of a rough quartz sample since the laser interferometer measuring device restrictions. Mist-abrasion machining method is proposed for corrective figuring of the optical materials. By the new figuring method, not only the polishing time decreases but also its efficiency increases by bigger abrasive grain size. The fundamental characteristics and its applicability for corrective figuring of a flat quartz sample are investigated. Flatness of the rough quartz sample is improved from PV=0.4μm to PV=0.1μm before the polishing process by mist-abrasion figuring.

Introduction

From the perspective of surface metrology, the surfaces machined through different machining procedures are of various defects. Problems such as roughness due to the sharp edges of tools, waviness resulting from machine vibrations, and chatter phenomena and defects in form due to misalignment of machine tools guideways are all inseparable parts of engineering surfaces. Each of these surface defects, depending on the applications of the part produced, can contribute in the malfunction of that part. For example, in optical parts, roughness of the surface leads to scattering of light, waviness results in the formation of shade or half-shade, any fault in the form causes a leak of unified focal point in lenses, and overall, a blurry picture look (1).

Numerous efforts have been made in order to obtain a perfect surface, but accessing an ideal surface sounds impossible. In this regard, despite all the advances in ultraprecision machining, achieving the precision in the nanometric form using ultraprecision grinding for the brittle pieces still seems to be difficult (2,3).

A different production process, in order to omit or minimize the problems mentioned above, has been implemented. In this respect, in order to improve the precision of the form, one can consider the corrective figuring. Corrective figuring is a costly and time consuming process. In this process, a definite contour is applied on the surface of the optical part. The contour could be part of a spherical, aspherical, or flat.

In the present article, in addition to represent different corrective figuring methods, mist-abrasion machining is also analyzed in order to remove the peak and valley on the flat optical part with high roughness, before polishing.

Among different methods for the local removal of matter, the following methods can be mentioned: polishing with small polishers, elastic emission machining (EEM), with ion beam machining (IBM), plasma assisted chemical vaporization machining (PCVM), magnetorheological finishing (MRF), and so on. Finishing with small size tools is difficult due to wearness because this abrasion leads to an inconsistency in the rate of material removal as well as problems in finishing large pieces. On the other hand, the EEM method, IBM, and PCVM require expensive and special equipment and MRF technique; considering costly and precise machinery is economically justifiable only in the mass production rate (2,4,5,6,7,8).

In this research, a corrective figuring method called mist-abrasion machining is suggested, which is similar to liquid honing, differing in that the abrasive slurry is sent to the nozzle through a low pressure pump. As illustrated in Fig.1, the abrasive particles of the slurry hit the surface of the workpiece through the jet of expelling compressed air leaving the exterior nozzle ring, upon leaving it. The amount and the way removed is done depend on the energy of liquid abrasive jet, properties of workpiece, as well as collision angle and stand of distance (2,4,6).

Among the main advantage of the mentioned method is that this way nozzle is situated at a distance from the workpiece; consequently, the accuracy of the method is distinguished from that of the machine accuracy. Also, the warping of the workpiece resulting from the champing forces has no effect on the accuracy of the machining method. Provided the above mentioned parameters are kept unchanged, the mode and material removal will always be constant, and for this reason, it is a suitable method for figure correction workpiece, but more pump wear would vanish. In the past researches, this method used to be applied on the polished optical part in which, in order to prevent the over roughness surface, microabrasive particle (WA No. 30000) was used in high pressure (2MPa, 4MPa, and 6MPa) (2). Fast wear of the pump and low speed of machining (inefficient material removal rate) are among the previous problems of the past method (2,4). In the new method, the idea of corrective figuring before the polishing process has been suggested.

In the next part of the paper, in addition to reviewing mist-abrasion machining, the process was simulated to predict the optimum conditions for the practical machining. In a high accuracy, the method was proved to be perfect, and the results of the experiment were used as proof of positive simulation results.

Mist-Abrasion Machining

Figure2 illustrates the schematic experimental apparatus used for mist-abrasion machining. Inside the machining chamber, the workpiece is mounted on a numerical control (NC) xy table. Slurry, consisting of 2% WA abrasive particles (WA No. 2000) mixed with pure water, is prepared. It is then conducted to the middle pipe in the nozzle with 1.2bar pressure. As shown in Fig.3, to achieve uniform slurry and to prevent the settling of the particle in the reservoir, a circulation loop has been used. The circuit has been designed in such a way to adjust the speed of the slurry, so that neither separation nor settlement of the particles would take place. The diameter of the nozzle used in this experiment is 0.5mm (Setojet 0.5 from Ikeucho Co. Japan (9)); the distance between the nozzle and the surface of the workpiece is 13.5mm and the collision angle is 90deg.

As illustrated in Fig. 1, the slurry is mixed with the air around the casing through the pipe inside the nozzle. The atomized slurry then accelerates toward the workpiece and upon collision with it, removes matter from the surface in nanometric scale. The volume of the matter removal is a function of kinetic energy of the abrasive (e=12mv2), as well as the frequency of collision. The frequency of collision (f) equals the number of the particle hitting the surface of the workpiece in the unit of time. This frequency can be calculated as follows: 
f=n=βwm=βρqm=βρAvm
1
where w is the mass flow rate of the slurry, q is volumetric flow rate of the slurry, A is the cross section of the nozzle, m and v are the mass and velocity of abrasive particles, respectively, when colliding the surface, ρ is the density of the abrasive in the slurry, and β is the constant coefficient when a liquid abrasive jet collides the surface.
The machining liquid contains certain rate percentage of abrasive particle (2wt%). So, at fixed outward jet flow and under ideal circumstances, multiplication of collision energy and its frequency for the particles with nonvariable mass will read as follows: 
E=ef=12mv2βρAvm=βρAv(32)
2
where E is the total collision energy. It is worth clarifying that in practice, the energy transferred to the workpiece is somewhat less than the theoretical one since some percentage of energy is wasted in the form of thermal energy.

To implement this method in corrective figuring, a number of experiments have been conducted on a quartz surface in different conditions. In order to determine the nozzle footprint, the workpiece is fixed with respect to the nozzle. The nozzle footprint has been measured two and three dimensionally using a cofocal laser measuring machine. For the footprint shown in Fig.4, all machining parameters are listed in Table 1. The footprint profile considering the appropriate design of the nozzle has a symmetrical outflow and is perfect for corrective figuring operation. Symmetric property of the footprint has led to the absence of any strive to perform a circular motion in order to achieve V- or U-shape symmetric footprint profiles, which is contrary to water jet abrasive machining (2). The U-shape symmetric footprint profile is used for corrective figuring of the surface and the tool path of corrective figuring shown in Fig.5.

The method of mist-abrasion machining suits corrective figuring better and that is due to V-shape symmetric profiles, which needs no circular motion and also speeds up the whole process by saving a lot of machining time

Corrective Figuring (Simulation and Experiment)

In order to correct the figure and eliminate any lump from the machined workpiece, error map of the machined surface is required, as shown in Fig.6. Due to technological limitations in implementing laser interferometer for a rough surface (surface roughness Ra=4μm), the capacitor method has been used to measure the surface roughness. The error map of the surface, which is the difference between the desired surface and the measured one as illustrated in Fig. 6, is stored as a two dimensional matrix in the simulating program. The flowchart for the corrective figuring program is shown in Fig.7.

According to the nozzle path, as shown in Fig. 5, the amount of material that is removed in every node is proportionate to the dwell time of the nozzle on the corresponding node. In this algorithm, the workpiece surface and the nozzle footprint are discretized. The dwell times of the nozzle on discretized points of the workpiece surface, which is termed as “dwell time matrix,” are calculated by using matrix algebra.

At first, in order to show the standard matrix footprint of the nozzle (as shown in Fig.8), the depth of material removal in a stationary state is measured by a noncontact laser measuring machine (Mitaka, NH-3N (10)) and saved as a matrix of standard nozzle footprints. The distance between nodes on the graph for standard nozzle footprints equals the nozzle pitch on the nozzle path (0.8mm) during corrective figuring, as shown in Fig. 5.

According to Preston law, the depth of matter removal is proportionate to the nozzle dwell time at the desired point, which can be stated as 
Zt=tt¯Zt¯
3
where Zt¯ is the matrix of the nozzle footprint for the standard dwell time, t¯ is the standard dwell time (3min according to machining conditions in Table 1), and Zt is the matrix depth of matter removal during the indefinite time t.
Since the radius of the nozzle footprint (approximately 4.8mm) is bigger than the machining pitch (0.8mm), the amount of the material removed at any point on the workpiece surface, according to superposition law, is, in fact, a combination of the material removed on the desired spot and other surrounding spots. As shown in Fig. 8, in order to simplify and calculate the amount of this material, first we consider the nozzle footprint as a single dimension, and will consequently have the following formula: 
Zt0(0)=t0t¯Zt¯(0)
4
 
Zt1(a)=t0t¯Zt¯(a)
5
 
Zti(ia)=t0t¯Zt¯(ia)
6
The sum of the material removed on the first node will be 
i+iZti(ia)=i+itit¯Zt¯(ia)
7
where i is the number of nodes from the center of the nozzle to the adjacent edge and a is the pitch distance in the x direction; by considering the direction of y, we will have 
j+ji+iZti,j(ia+ja)=j+ji+iti,jt¯Zt¯(ia+jb)
8
where b is the pitch distance in the y direction. With regard to the fact that the depth of material removal at any point equals the difference in height between the measured surface and the required one at the same spot, we will have 
D=SW
9
where S is the required surface and W represents the measured surface before figure correction. In order to correct the form and eliminate the discrepancy, the formula should read as 
D(x0,y0)=j+ji+iti,jt¯Zt¯(ia+jb)
10
Considering no severe changes on the surface, it can be assumed that D is the same in points (Xi,Yi) and (Xo,Yo). In other words, the dwell times t(Xi,Yi) and t(Xo,Yo) will also be equal and therefore 
ti,j=D(xi,yi)t¯j+ji+iZt¯(ia+jb)
11
The denominator is shown as D¯, which is equal to the total material removed at the standard time, or to put it another way, D¯ equals the whole matrix arrangement of the nozzle footprint. 
ti,j=D(xi,yi)t¯D¯
12
where tij is the first approximate dwell time matrix. By this dwell time matrix, we can find the first iteration surface topography, as it is shown S in Fig.9. S and S are different due to the approximation in calculation; therefore, (SS) results in various positive or negative error outcomes in different nodes.

In the next step, while identifying these spots, the dwell time matrix arrays for the relevant spots are decreased or increased a little for positive and negative errors, respectively. By the iterative procedure, desirable PV value is achieved and the resulting surface is simulated in the end. Finally, the program for the relevant figure correction is printed as an NC program.

The corrective figuring of a number of samples has been done by the proposed method. The condition of the corrective figuring process is determined by the simulation program. The surface topography of a sample before and after corrective figuring and the result of simulation program are shown in Figs.10a,10b,10c, respectively. As shown in the figure, the out of flatness of the surface decreased from PV=0.4μm to PV=0.1μm, without any change in surface roughness value. The deviation of simulation and experimental result is due to nonconsistency of the nozzle footprint and some practical restrictions in actual machining.

Conclusion

In this research, unlike the conventional figuring techniques (2,7,8), a figure correction method was introduced in order to figure an optical piece before polishing process in which its out of flatness is proportionate to surface roughness. By the proposed corrective figuring method, the surface form error and waviness are decreased. Therefore, the next polishing process time will decrease significantly due to remove just surface roughness in the polishing stage. Furthermore, in “mist-abrasion machining,” unlike “nanoabrasive machining” (2), the need for circular motion is eliminated, and also by the use of bigger abrasive particles, the efficiency of figuring is increased. The absence of high pressure pumps, which have high rate of wear and tear, is another advantage of the proposed method. As shown in Fig. 10, in this recommended method, through determining the conditions of process by means of a simulation program and relevant correction, without any change in surface roughness value, the surface fault decreased from PV=0.4μm to PV=0.1μm.

Acknowledgment

I would like to extend my gratitude to all members of the Precision Machine Laboratory in Toyohashi University of Technology, in Japan, as well as the staff of production workshop in the same university.

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