- Email: [email protected]
A contact inspection system for aspheric optical components
Optik 127 (2016) 7572–7577
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
A contact inspection system for aspheric optical components Gufeng Qiu a,b , Xudong Cui b,∗ a
Institute of Modern Optics, Soochow University, Suzhou, Jiangsu, P.R. China, 215006, PR China Science and Technology on Plasma Physics Laboratory, Research Center of Laser Fusion, CAEP, Mianyang, Sichuan, P.R. China, 621900, PR China b
a r t i c l e
i n f o
Article history: Received 27 May 2015 Accepted 26 May 2016 Keywords: Geometric optics Optical measurement techniques Optical inspection Aspheric
a b s t r a c t We construct a contact inspection system and realize low contact force and high precision measurements for aspheric optical components. The inspection and calibration principles, as well as the reliability of the system are thoroughly studied. Our investigations show that the system could provide low contact force measurement (low than 0.7 mN), fast scanning speed (2 mm/s), and high accuracy (as low as 0.5 m), being a good option to the contact inspections of aspheric optical components. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction The production rate of aspheric optical components depends strongly on the inspection techniques [1]. For the contact inspection systems (CISs), the contact force, measurement speed, precision are key specifications. Currently, only few convenient CISs are available for aspheric, i.e., the Form Talysurf® (the precision is 0.3 m, minima contact force is 0.7–1 mN, scanning speed is 0.25–2 mm/s) [2] and Panasonic UA3P® (i.e., for UA3P-4, the precision is 0.05 m, contact force is 0.15–0.3 mN, scanning speed is 0.01–10 mm/s), the coordinate measuring machine (CMM) (CMM measurement speed is 4–5s/dot, precision is 3–5 m, contact force is 0.1–0.3 N) [3–5], and the length gauge profiles (LGP) (minima contact force is 0.75N, precision is 1–2 m, speed even more slow than other two systems) [5] [6]. Typically, for industrial applications, CMM and LGP can hardly meet the requirements of high precision measurements, while other high precision systems (i.e.,UA3 P or Form Talysurf) are still with high costs. A low cost system with desired specifications (high precision, low contact force and fast inspection speed) for aspheric fabrication is therefore strongly preferred [7]. In this work, using the lever principle, we realized a contact inspection system for aspheric with very low contact force, fast scanning speed and high measurement precision. Our initial studies show that this system is feasible and might have huge application potentials in this area. 2. Principles The principles of our system are shown in Fig. 1a. It consists of a probe (made by ruby, the probe is fixed on a rail so that it can move along the horizontal direction); a measuring pole (with an arc optical grating); an adjusting nut for the contact force between the probe and testing component; an air bearing and a grating reader. The fabricated system is shown in Fig. 1b.
∗ Corresponding author. E-mail address: [email protected] (X. Cui). http://dx.doi.org/10.1016/j.ijleo.2016.05.123 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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Fig. 1. (a) Schematic drawing of the system; (b) The fabricated system.
Fig. 2. The coordinate system used for the mathematical model.
The measurement procedures are as follows: (1) at the beginning, by moving the horizontal rail, let the probe apex (needle) contact with the left edge of the component under testing (Fig. 1a and b); (2) Then, we start to determine the center of testing component, by tuning the adjusting gear on the platform and making the reading of apex minima (convex) or maximum (concave); Note that this step should be repeated until the center is found;(3) As long as the center and left edge are determined, we start to move the horizontal rail toward the right direction at a constant speed, until the probe needle is moved to the rightmost side of the component. During the movement of the probe, a data collection card with the function of synchronous latch is used to simultaneously record the current position of rail xi and the reading of measuring grating zi . Note that the zero position of the grating is defined as the center of the arc surface. Therefore, the reading data (xi ,zi ) includes the profile information of the component under testing. 3. Measuring model A coordinate system shown in Fig. 2 is employed to set up the measuring model. The original point is selected as the central rotating point of the air bearing when the horizontal rail is located at the zero point; the Z axis is defined as the line that connecting the central rotating point of air bearing and the zero point of arc grating surface (Fig. 2); X axis is along the right direction as indicated in the Figure. When the grating reading is zi , the probe rotates an angle of i = zi /r1 , where r1 is
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the radius of the arc grating surface. We assume the position of the rail is xi . So the central position of the probe (xi , zi ) can be expressed as:
xi = xi + r2 cos(/2 + − i )
(1)
zi = r2 sin(/2 + − i )
(2)
where is the angle between the line connecting the probe center and the center of air bearing and the line of measuring pole (Z axis); r2 is the distance between the probe center and the center of air bearing. Once the radius of the probe is known, we can analyze the profile error of the component. Note that above analysis is under ideal conditions. In practice, how to precisely determine the three parameters , r1 and r2 in the model is the key to the application of our system. We then must take following two aspects into considerations: (1) There exist discrepancies between the theoretical design values and the real measurement values. So a calibrating model is needed to minimize this discrepancy; (2) In the measurement, what we obtained is the central traces of the probe apex. Since the probe is not an ideal point but more like a standard ball with a certain radius, a dada analysis model is then required to convert these traces into the practical aspheric traces. 4. Calibrating model Bearing these in minds, First, we then use a standard spherical surface with known radius R to determine the three key parameters , r1 and r2 . The least square method is employed to analyze the measured data; the minima mean square root values of , r1 and r2 for the standard spherical surface will be used as the realistic parameters. Suppose the measured dot matrix data are{(xi , zi )}, the theoretical design parameters are , r1 and r2 , the real parameters are , r1 and r2 , and
= + d
(3)
r1 = r1 + dr1
(4)
r2 = r2 + dr2
(5) ,
During the whole measurement, the traces of the probe center {(xi zi
xi = xi + (r2 + dr2 ) cos(/2 + + d − i )
)}
can be expressed as: (6)
zi = (r2 + dr2 ) sin(/2 + + d − i )
(7)
where i = zi /(r1 + dr1 ). Assuming the radius of the probe apex is r, then the measured traces of the probe center should be R + r. Considering the placement error of aspheric component, we suppose that in above coordinate system, the spherical center of the standard sphere is located at (x0 , z0 ). We then define the function: F(x0 , z0 , d, dr1 , dr2 ) =
n
(
2
2
(xi − x0 ) + (zi − z0 ) − (R + r))
2
(8)
i=0
In ideal cases, the function F should be zero. Therefore, in our work we then make the function F be zero with corresponding variations d, dr1 , dr2 , so that the realistic parameters can be determined. Based on the principle of least square method, F is minima only that the partial derivatives of corresponding variations are 0. Namely, the conditions are:
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂x0
(9)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂z0
(10)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂d
(11)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂dr1
(12)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂dr2
(13)
By solving Eqs. (9)–(13), d, dr1 , dr2 are obtained and then , r1 r2 can be determined with Eqs. (3)–(5). The system calibration can then be finished.
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5. Data analysis model As we described in previous section, the results obtained by our inspection system are the moving traces of the probe apex center, rather than the practical profiles of aspheric. In practice, it is difficult to do so and we turn to find the deviations between the practical values and theoretical designs. Since the radius of the probe apex is relatively small compared to the curvature of aspheric component, we can use the deviations between the practical traces of moving probe to replace the deviations between practical aspheric and theoretical designs. Based on the above analysis, we can have the data analysis model for our system as follows. Suppose the equation for the aspheric surface is z = f(x), (x0 , f(x0 )) is the point on the aspheric surface. Then, when the probe apex with radius r contacting with the aspheric surface, the position of probe apex (x, z) can be expressed as: x = x0 −
z = f (x0 ) +
rf (x0 )
(14)
1 + f (x0 )2
r
(15)
1 + f (x0 )2
In the bus line analysis of aspheric surface, we mainly consider the placement error of aspheric surface and the deviation of axis. Assuming the top of measured aspheric surface in the coordinate system is (xzero , zzero ), and the angle between the axis of aspheric surface and the Z axis is ␣. For the measured traces of probe center{(xi , zi )}, we can use the following coordinate transformation relations to convert it into the local coordinate system of aspheric standard equations.
xi = (xi − xzero ) cos(˛) − (zi − zzero ) sin(˛)
(16)
zi = (xi − xzero ) sin(˛) + (zi − zzero ) cos(˛) ,
(17)
)}is
Therefore, the determined traces {(xi zi the measured traces of probe centers by Eqs. (16) and (17). For a given (xi , zi ), one can obtain corresponding xi0 by Eq. (14), so that the theoretical values for probe center corresponding to the aspheric surface can be obtained by Eq. (15). In principle, the measured data error of aspheric surface can be expressed as:
zie = zi − f (xi0 ) +
r
(18)
1 + f (xi0 )2
Once the parameters xzero , zzero , ␣ are determined, one can have the error data. Similarly, by solving the minima mean square root of these parameters x zero , z zero , ˛ between measured and theoretical values, we can approximately get the real parameters xzero , zzero , ␣. We define a function: F(xzero , zzero , ˛) =
n
(zi − f (xi0 ) +
i=0
2
r
1 + f (xi0 )
2
)
(19)
By solving following equations
∂F(xzero , zzero , ˛) =0 ∂xzero
(20)
∂F(xzero , zzero , ˛) =0 ∂zzero
(21)
∂F(xzero , zzero , ˛) =0 ∂˛
(22)
One can get x zero , z zero , ˛ . The obtained parameters x zero , z zero , ˛ , combined with Eq. (18), we then have the error data for the measured aspheric surfaces. 6. Experimental results and discussions To validate the feasibility of our system and models, we build such an inspection equipment (Fig. 1b). The horizontal rail is the air floating rail, and the parameters for the air floating reader are as follows: r1 = 62.5 mm, r2 = 82.5 mm, =1.865 rad, the radius of probe is r = 1.25 mm. The grating used here is the RENISHAW grating with a solution of 0.1 m. The radius of the standard ball is 51.85 mm. Through the calibrating methods shown in previous section, we obtained the real parameters of the standard ball are: r1 = 62.33050 mm, r2 = 81.43922 mm, =1.847037 rad, well in the acceptable range with high precision. After calibration, we measured a standard plane and another standard spherical surface with a curvature of −22.5060 mm−1 to check the consistency and repeatability of our system. Table 1 lists the 10 sets of measured data for the standard plane. The maximum PV is 0.17 m, the maximum RMS is 21.6 nm, indicating that the consistency error of our device is less than 0.17 m (within the range of 120 mm aperture). While for the standard spherical surface, we repeat
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Table 1 Testing results for the standard plane. Testing Numbers
1
2
3
4
5
PV(m) RMS(nm)
0.16 20
0.15 19.8
0.16 16.8
0.15 17.9
0.13 17.5
Testing Numbers
6
7
8
9
10
PV(m) RMS(nm)
0.14 18.9
0.14 16.6
0.16 19.8
0.17 21.6
0.14 17.7
Table 2 Measured results for a standard spherical surface under different scanning speed. Speed(mm/s)
0.25
0.5
0.75
1.0
PV(m) RMS(nm)
0.36 64.7
0.39 60.5
0.30 50.1
0.4 67.2
Speed(mm/s) PV(m) RMS(nm)
1.25 0.21 26.7
1.5 0.23 41.5
1.75 0.38 70.1
2.0 0.45 95
Fig. 3. Measured results for a standard plane. The horizontal axis represents the aperture of testing samples, the vertical axis stands for the deviation from theoretical values.
the measurements under different testing speeds from 0.25 mm/s to 2.0 mm/s, at an interval of 0.25 mm/s. The results are listed in Table 2. We can see that even under different scanning speed, all the results are with very good consistency: the maximum PV is less than 0.45 m and the maximum RMS is close to 95 nm. For the standard plane (surface) the errors can be regarded as zero (as true value or reference), and hence, the testing results reflect the accuracy of our instrument itself. As a consequence, we can safely conclude that the accuracy of our instrument is well below 0.5 m. Fig. 3 and Fig. 4 show one of the resulting curves after experiments for the standard plane and the spherical surface, respectively. The horizontal axis represents the aperture of testing samples, the vertical axis stands for the deviation from theoretical values. We have to stress here that, for our device, the measurement accuracy can be well controlled by adjusting the length ratio of the swing arm (r1 ) to the measuring arm (r2 ). For example, if we assume the measurement accuracy is 0.5 m when r1 = r2 ; when r1 = 2r2 , the accuracy can reach 0.25 m. When different lengths of measuring arm (r2 ) are configured, one can realize measurements with different accuracy. Note that in this case, the measurement range will also change. For higher accuracy (i.e., less than 0.1 m), more work need to be done with our configurations. In order to apply our instrument to the measurement of aspheric surface, we test a real aspheric surface and compare the results with that of from the Talysurf® measurement. The parameters of aspheric surface are as follows (units: mm): R = −20.15, K = −1.035356, a4 = −5.685242e-6, a6 = 7.934483e-10, a8 = 3.635248e-12, a10 = −5.883743e-16. The results are shown in Fig. 5. The blue curve represents the result from the TaylorSurf® measurement, while the green curve is our measurement. Two results have very good agreement with each other, showing that our instrument has comparable accuracy with commercial apparatus. Some data errors are found in the curve and we thought this originated from the displacement error during measurement on different platforms.
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Fig. 4. Measured results for a calibrated standard spherical surface. The horizontal axis represents the aperture of testing samples, the vertical axis stands for the deviation from theoretical values.
Fig. 5. Measured results for a true aspheric surface. Blue curve is obtained from the TaylorSurf® and Green curve is from our instrument in this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
In addition, the contact force can be easily changed by adjusting the nut (located at the right side of the air bearing in Fig. 1a and b) in our system and the force is measured with a digital scale. Force as low as 0.7 mN is obtained in the experiments. In principle, the contact force can be adjusted to be zero, and hence we can even have more less contact force in practice. 7. Conclusion In summary, to realize fast, low contact force and high precise inspection for aspheric surfaces, we design an equipment for such purposes, based on the lever principle. We build such an inspection system and the feasibility of the system is validated. The calibration and data analysis model are also given. Our investigations show that, with our system, the measured accuracy can be as low as 0.5 m, the contact force is lower than 0.7 mN, and the scanning speed is higher than 2 mm/s. Note that although the measured model in this work is only for the bus line inspection of aspheric surface, it is very easy to extend to the 3D measurement by simply adding an air floating rotating stage to realize high precision 3D profile inspection (like UA3P). We believe that our designs (low contact force, fast, high precision and low cost) would provide a new way to the inspection of aspheric surfaces, and further improve the fabrication efficiency and lower the costs. References [1] [2] [3] [4] [5] [6] [7]
D. Malacara, Optical Shop Testing, 3rd ed., Wiley, 2007. S. Gomez, K. Hale, J. Burrows, B. Griffiths, Meas. Sci. Technol. 9 (1998) 607. A. Wozniak, J. Mayer, Meas. Sci. Technol. 23 (2012) 025001. A. Muralikrishnan, J. Stone, C. Shakarji, J. Stoup, Meas. Sci. Technol. 23 (2012) 025002. K. Fan, F. Cheng, W. Wang, Y. Chen, J. Lin, Meas. Sci. Technol. 21 (2010) 054002. Z. Dong, H. Cheng, X. Ye, H. Tam, Appl. Opt. 53 (2014) 4997. V.A. Gorshkov, N.G. Korneev, Y.B. Lyamin, A.Y. Papaev, A.V. Podobryanski, J. Opt. Technol. 70 (2003) 724.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
A contact inspection system for aspheric optical components Gufeng Qiu a,b , Xudong Cui b,∗ a
Institute of Modern Optics, Soochow University, Suzhou, Jiangsu, P.R. China, 215006, PR China Science and Technology on Plasma Physics Laboratory, Research Center of Laser Fusion, CAEP, Mianyang, Sichuan, P.R. China, 621900, PR China b
a r t i c l e
i n f o
Article history: Received 27 May 2015 Accepted 26 May 2016 Keywords: Geometric optics Optical measurement techniques Optical inspection Aspheric
a b s t r a c t We construct a contact inspection system and realize low contact force and high precision measurements for aspheric optical components. The inspection and calibration principles, as well as the reliability of the system are thoroughly studied. Our investigations show that the system could provide low contact force measurement (low than 0.7 mN), fast scanning speed (2 mm/s), and high accuracy (as low as 0.5 m), being a good option to the contact inspections of aspheric optical components. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction The production rate of aspheric optical components depends strongly on the inspection techniques [1]. For the contact inspection systems (CISs), the contact force, measurement speed, precision are key specifications. Currently, only few convenient CISs are available for aspheric, i.e., the Form Talysurf® (the precision is 0.3 m, minima contact force is 0.7–1 mN, scanning speed is 0.25–2 mm/s) [2] and Panasonic UA3P® (i.e., for UA3P-4, the precision is 0.05 m, contact force is 0.15–0.3 mN, scanning speed is 0.01–10 mm/s), the coordinate measuring machine (CMM) (CMM measurement speed is 4–5s/dot, precision is 3–5 m, contact force is 0.1–0.3 N) [3–5], and the length gauge profiles (LGP) (minima contact force is 0.75N, precision is 1–2 m, speed even more slow than other two systems) [5] [6]. Typically, for industrial applications, CMM and LGP can hardly meet the requirements of high precision measurements, while other high precision systems (i.e.,UA3 P or Form Talysurf) are still with high costs. A low cost system with desired specifications (high precision, low contact force and fast inspection speed) for aspheric fabrication is therefore strongly preferred [7]. In this work, using the lever principle, we realized a contact inspection system for aspheric with very low contact force, fast scanning speed and high measurement precision. Our initial studies show that this system is feasible and might have huge application potentials in this area. 2. Principles The principles of our system are shown in Fig. 1a. It consists of a probe (made by ruby, the probe is fixed on a rail so that it can move along the horizontal direction); a measuring pole (with an arc optical grating); an adjusting nut for the contact force between the probe and testing component; an air bearing and a grating reader. The fabricated system is shown in Fig. 1b.
∗ Corresponding author. E-mail address: [email protected] (X. Cui). http://dx.doi.org/10.1016/j.ijleo.2016.05.123 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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Fig. 1. (a) Schematic drawing of the system; (b) The fabricated system.
Fig. 2. The coordinate system used for the mathematical model.
The measurement procedures are as follows: (1) at the beginning, by moving the horizontal rail, let the probe apex (needle) contact with the left edge of the component under testing (Fig. 1a and b); (2) Then, we start to determine the center of testing component, by tuning the adjusting gear on the platform and making the reading of apex minima (convex) or maximum (concave); Note that this step should be repeated until the center is found;(3) As long as the center and left edge are determined, we start to move the horizontal rail toward the right direction at a constant speed, until the probe needle is moved to the rightmost side of the component. During the movement of the probe, a data collection card with the function of synchronous latch is used to simultaneously record the current position of rail xi and the reading of measuring grating zi . Note that the zero position of the grating is defined as the center of the arc surface. Therefore, the reading data (xi ,zi ) includes the profile information of the component under testing. 3. Measuring model A coordinate system shown in Fig. 2 is employed to set up the measuring model. The original point is selected as the central rotating point of the air bearing when the horizontal rail is located at the zero point; the Z axis is defined as the line that connecting the central rotating point of air bearing and the zero point of arc grating surface (Fig. 2); X axis is along the right direction as indicated in the Figure. When the grating reading is zi , the probe rotates an angle of i = zi /r1 , where r1 is
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the radius of the arc grating surface. We assume the position of the rail is xi . So the central position of the probe (xi , zi ) can be expressed as:
xi = xi + r2 cos(/2 + − i )
(1)
zi = r2 sin(/2 + − i )
(2)
where is the angle between the line connecting the probe center and the center of air bearing and the line of measuring pole (Z axis); r2 is the distance between the probe center and the center of air bearing. Once the radius of the probe is known, we can analyze the profile error of the component. Note that above analysis is under ideal conditions. In practice, how to precisely determine the three parameters , r1 and r2 in the model is the key to the application of our system. We then must take following two aspects into considerations: (1) There exist discrepancies between the theoretical design values and the real measurement values. So a calibrating model is needed to minimize this discrepancy; (2) In the measurement, what we obtained is the central traces of the probe apex. Since the probe is not an ideal point but more like a standard ball with a certain radius, a dada analysis model is then required to convert these traces into the practical aspheric traces. 4. Calibrating model Bearing these in minds, First, we then use a standard spherical surface with known radius R to determine the three key parameters , r1 and r2 . The least square method is employed to analyze the measured data; the minima mean square root values of , r1 and r2 for the standard spherical surface will be used as the realistic parameters. Suppose the measured dot matrix data are{(xi , zi )}, the theoretical design parameters are , r1 and r2 , the real parameters are , r1 and r2 , and
= + d
(3)
r1 = r1 + dr1
(4)
r2 = r2 + dr2
(5) ,
During the whole measurement, the traces of the probe center {(xi zi
xi = xi + (r2 + dr2 ) cos(/2 + + d − i )
)}
can be expressed as: (6)
zi = (r2 + dr2 ) sin(/2 + + d − i )
(7)
where i = zi /(r1 + dr1 ). Assuming the radius of the probe apex is r, then the measured traces of the probe center should be R + r. Considering the placement error of aspheric component, we suppose that in above coordinate system, the spherical center of the standard sphere is located at (x0 , z0 ). We then define the function: F(x0 , z0 , d, dr1 , dr2 ) =
n
(
2
2
(xi − x0 ) + (zi − z0 ) − (R + r))
2
(8)
i=0
In ideal cases, the function F should be zero. Therefore, in our work we then make the function F be zero with corresponding variations d, dr1 , dr2 , so that the realistic parameters can be determined. Based on the principle of least square method, F is minima only that the partial derivatives of corresponding variations are 0. Namely, the conditions are:
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂x0
(9)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂z0
(10)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂d
(11)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂dr1
(12)
∂F(x0 , z0 , d, dr1 , dr2 ) =0 ∂dr2
(13)
By solving Eqs. (9)–(13), d, dr1 , dr2 are obtained and then , r1 r2 can be determined with Eqs. (3)–(5). The system calibration can then be finished.
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5. Data analysis model As we described in previous section, the results obtained by our inspection system are the moving traces of the probe apex center, rather than the practical profiles of aspheric. In practice, it is difficult to do so and we turn to find the deviations between the practical values and theoretical designs. Since the radius of the probe apex is relatively small compared to the curvature of aspheric component, we can use the deviations between the practical traces of moving probe to replace the deviations between practical aspheric and theoretical designs. Based on the above analysis, we can have the data analysis model for our system as follows. Suppose the equation for the aspheric surface is z = f(x), (x0 , f(x0 )) is the point on the aspheric surface. Then, when the probe apex with radius r contacting with the aspheric surface, the position of probe apex (x, z) can be expressed as: x = x0 −
z = f (x0 ) +
rf (x0 )
(14)
1 + f (x0 )2
r
(15)
1 + f (x0 )2
In the bus line analysis of aspheric surface, we mainly consider the placement error of aspheric surface and the deviation of axis. Assuming the top of measured aspheric surface in the coordinate system is (xzero , zzero ), and the angle between the axis of aspheric surface and the Z axis is ␣. For the measured traces of probe center{(xi , zi )}, we can use the following coordinate transformation relations to convert it into the local coordinate system of aspheric standard equations.
xi = (xi − xzero ) cos(˛) − (zi − zzero ) sin(˛)
(16)
zi = (xi − xzero ) sin(˛) + (zi − zzero ) cos(˛) ,
(17)
)}is
Therefore, the determined traces {(xi zi the measured traces of probe centers by Eqs. (16) and (17). For a given (xi , zi ), one can obtain corresponding xi0 by Eq. (14), so that the theoretical values for probe center corresponding to the aspheric surface can be obtained by Eq. (15). In principle, the measured data error of aspheric surface can be expressed as:
zie = zi − f (xi0 ) +
r
(18)
1 + f (xi0 )2
Once the parameters xzero , zzero , ␣ are determined, one can have the error data. Similarly, by solving the minima mean square root of these parameters x zero , z zero , ˛ between measured and theoretical values, we can approximately get the real parameters xzero , zzero , ␣. We define a function: F(xzero , zzero , ˛) =
n
(zi − f (xi0 ) +
i=0
2
r
1 + f (xi0 )
2
)
(19)
By solving following equations
∂F(xzero , zzero , ˛) =0 ∂xzero
(20)
∂F(xzero , zzero , ˛) =0 ∂zzero
(21)
∂F(xzero , zzero , ˛) =0 ∂˛
(22)
One can get x zero , z zero , ˛ . The obtained parameters x zero , z zero , ˛ , combined with Eq. (18), we then have the error data for the measured aspheric surfaces. 6. Experimental results and discussions To validate the feasibility of our system and models, we build such an inspection equipment (Fig. 1b). The horizontal rail is the air floating rail, and the parameters for the air floating reader are as follows: r1 = 62.5 mm, r2 = 82.5 mm, =1.865 rad, the radius of probe is r = 1.25 mm. The grating used here is the RENISHAW grating with a solution of 0.1 m. The radius of the standard ball is 51.85 mm. Through the calibrating methods shown in previous section, we obtained the real parameters of the standard ball are: r1 = 62.33050 mm, r2 = 81.43922 mm, =1.847037 rad, well in the acceptable range with high precision. After calibration, we measured a standard plane and another standard spherical surface with a curvature of −22.5060 mm−1 to check the consistency and repeatability of our system. Table 1 lists the 10 sets of measured data for the standard plane. The maximum PV is 0.17 m, the maximum RMS is 21.6 nm, indicating that the consistency error of our device is less than 0.17 m (within the range of 120 mm aperture). While for the standard spherical surface, we repeat
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Table 1 Testing results for the standard plane. Testing Numbers
1
2
3
4
5
PV(m) RMS(nm)
0.16 20
0.15 19.8
0.16 16.8
0.15 17.9
0.13 17.5
Testing Numbers
6
7
8
9
10
PV(m) RMS(nm)
0.14 18.9
0.14 16.6
0.16 19.8
0.17 21.6
0.14 17.7
Table 2 Measured results for a standard spherical surface under different scanning speed. Speed(mm/s)
0.25
0.5
0.75
1.0
PV(m) RMS(nm)
0.36 64.7
0.39 60.5
0.30 50.1
0.4 67.2
Speed(mm/s) PV(m) RMS(nm)
1.25 0.21 26.7
1.5 0.23 41.5
1.75 0.38 70.1
2.0 0.45 95
Fig. 3. Measured results for a standard plane. The horizontal axis represents the aperture of testing samples, the vertical axis stands for the deviation from theoretical values.
the measurements under different testing speeds from 0.25 mm/s to 2.0 mm/s, at an interval of 0.25 mm/s. The results are listed in Table 2. We can see that even under different scanning speed, all the results are with very good consistency: the maximum PV is less than 0.45 m and the maximum RMS is close to 95 nm. For the standard plane (surface) the errors can be regarded as zero (as true value or reference), and hence, the testing results reflect the accuracy of our instrument itself. As a consequence, we can safely conclude that the accuracy of our instrument is well below 0.5 m. Fig. 3 and Fig. 4 show one of the resulting curves after experiments for the standard plane and the spherical surface, respectively. The horizontal axis represents the aperture of testing samples, the vertical axis stands for the deviation from theoretical values. We have to stress here that, for our device, the measurement accuracy can be well controlled by adjusting the length ratio of the swing arm (r1 ) to the measuring arm (r2 ). For example, if we assume the measurement accuracy is 0.5 m when r1 = r2 ; when r1 = 2r2 , the accuracy can reach 0.25 m. When different lengths of measuring arm (r2 ) are configured, one can realize measurements with different accuracy. Note that in this case, the measurement range will also change. For higher accuracy (i.e., less than 0.1 m), more work need to be done with our configurations. In order to apply our instrument to the measurement of aspheric surface, we test a real aspheric surface and compare the results with that of from the Talysurf® measurement. The parameters of aspheric surface are as follows (units: mm): R = −20.15, K = −1.035356, a4 = −5.685242e-6, a6 = 7.934483e-10, a8 = 3.635248e-12, a10 = −5.883743e-16. The results are shown in Fig. 5. The blue curve represents the result from the TaylorSurf® measurement, while the green curve is our measurement. Two results have very good agreement with each other, showing that our instrument has comparable accuracy with commercial apparatus. Some data errors are found in the curve and we thought this originated from the displacement error during measurement on different platforms.
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Fig. 4. Measured results for a calibrated standard spherical surface. The horizontal axis represents the aperture of testing samples, the vertical axis stands for the deviation from theoretical values.
Fig. 5. Measured results for a true aspheric surface. Blue curve is obtained from the TaylorSurf® and Green curve is from our instrument in this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
In addition, the contact force can be easily changed by adjusting the nut (located at the right side of the air bearing in Fig. 1a and b) in our system and the force is measured with a digital scale. Force as low as 0.7 mN is obtained in the experiments. In principle, the contact force can be adjusted to be zero, and hence we can even have more less contact force in practice. 7. Conclusion In summary, to realize fast, low contact force and high precise inspection for aspheric surfaces, we design an equipment for such purposes, based on the lever principle. We build such an inspection system and the feasibility of the system is validated. The calibration and data analysis model are also given. Our investigations show that, with our system, the measured accuracy can be as low as 0.5 m, the contact force is lower than 0.7 mN, and the scanning speed is higher than 2 mm/s. Note that although the measured model in this work is only for the bus line inspection of aspheric surface, it is very easy to extend to the 3D measurement by simply adding an air floating rotating stage to realize high precision 3D profile inspection (like UA3P). We believe that our designs (low contact force, fast, high precision and low cost) would provide a new way to the inspection of aspheric surfaces, and further improve the fabrication efficiency and lower the costs. References [1] [2] [3] [4] [5] [6] [7]
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