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Calibration of the straightness and orthogonality error of a laser feedback high-precision stage using self-calibration methods
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Measurement Science and Technology Meas. Sci. Technol. 25 (2014) 125007 (10pp)
doi:10.1088/0957-0233/25/12/125007
Calibration of the straightness and orthogonality error of a laser feedback high-precision stage using self-calibration methods Dongmin Kim1,4, Kihyun Kim2,4, Sang Hyun Park3 and Sangdon Jang3 1
Korea Research Institute of Standards and Science (KRISS), Daejeon, Korea 305–340 Korea Intellectual Property Office, Daejeon, Korea 302–784 3 Mechatronics and Manufacturing Technology Center, Samsung Electronics, Suwon, Korea 443–742 2
E-mail:
[email protected] and
[email protected] Received 29 May 2014, revised 31 August 2014 Accepted for publication 15 October 2014 Published 14 November 2014 Abstract
An ultra high-precision 3-DOF air-bearing stage is developed and calibrated in this study. The stage was developed for the transportation of a glass or wafer with x and y following errors in the nanometer regime. To apply the proposed stage to display or semiconductor fabrication equipment, x and y straightness errors should be at the sub-micron level and the x–y orthogonality error should be in the region of several arcseconds with strokes of several hundreds of mm. Our system was designed to move a 400 mm stroke on the x axis and a 700 mm stroke on the y axis. To do this, 1000 mm and 550 mm bar-type mirrors were adopted for real time Δx and Δy laser measurements and feedback control. In this system, with the laser wavelength variation and instability being kept to a minimum through environmental control, the straightness and orthogonality become purely dependent upon the surface shape of the bar mirrors. Compensation for the distortion of the bar mirrors is accomplished using a self-calibration method. The successful application of the method nearly eliminated the straightness and orthogonality errors of the stage, allowing their specifications to be fully satisfied. As a result, the straightness and orthogonality errors of the stage were successfully decreased from 4.4 μm to 0.8 μm and from 0.04° to 2.48 arcsec, respectively. Keywords: calibration, self-calibration, laser interferometer, bar mirror, straightness, orthogonality (Some figures may appear in colour only in the online journal)
1. Introduction
There are several calibration methods. The first method is known as parametric calibration [1]. This method uses matrices to transform an array of 6-DOF errors at the point of measurement to those at the point of interest. In this case, the level of accuracy of the errors becomes highly dependent on the level of accuracy of the kinematic model as well as the level of measurement accuracy of the 6-DOF errors. The second method uses standard specimens. This is most likely the most common, accurate and convenient method. However, it can be expensive and even inaccurate as the calibration range becomes larger because a specimen that exceeds
In a laser feedback stage system, motion of the stage is purely determined by its bar mirror shape. A 1D error caused either by the x or y bar mirror shape is referred to as a straightness error, while a 2D error caused by the out-of-squareness of the x and y bar mirror is referred to as an orthogonality error in this paper. The objective of the research is to calibrate and compensate for these errors. 4
Authors to whom any correspondence should be addressed.
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the straightness profile of a straightedge. In setup 1, the sensor signal represents the stage straightness error, M(y), plus the straightedge error, S(y), during the motion of the stage. In setup 2, the straightedge is reversed such that the sensor signal represents the difference between M(y) and S(y). Solving these two equations for M(y) and S(y) gives equation (2).
I1 (y ) = S (y ) + M (y ) I2 (y ) = S (y ) − M (y )
(1)
I1 (y ) + I2 (y ) 2 I1 (y ) − I2 (y ) M (y ) = 2
(2)
Figure 1. The self-calibration of the straightness of a bar mirror.
S (y ) =
2.2. Self-calibration of orthogonality
commercially available dimensions should be custom-made and the cost is linked to the level of accuracy required. The third method is the self-calibration method. The basic idea and benefit of this method is that even low-quality specimens can be employed in the calibration because their errors are separated from the errors of interest. For this reason, the self-calibration method is the most cost-effective method for large-range calibrations. Also, the level of achievable accuracy can reach the level of system repeatability if the process is executed well. The self-calibration method, also known as the reversal method or error separation technique in the literature, has been applied by several authors with slightly different approaches for the calibration of single axis straightness errors [2] and orthogonality errors [3]. Lieberman [4] first introduced the concept of rotating a specimen in an earlier e-beam lithography system calibration. This 2D calibration was then mathematically organized by Raugh [5, 6] and Takac [7]. Later, Ye [8] presented an algorithm that dealt with 2D systematic errors. However, this 2D calibration approach was applicable only for square stroke systems. For the calibration of a laser feedback stage, Ruijl [9] applied an error separation technique for the straightness calibration of a mirror and Hume’s angular measuring method [10] was used for orthogonality calibrations. In this paper, the methods of Evans [2] and Ruijl [9] are applied for the calibration and compensation of a high-precision, laser feedback stage. Section 2 briefly introduces this method with a few key formulas. Detailed analytics stem from an earlier study [9]. The actual setups and calibration procedures for straightness and orthogonality calibrations are presented in sections 4 and 5, respectively.
Figure 2 shows the concept and sequence of the self-calibration of bar mirror orthogonality. A polygon is attached to a moving stage and its two orthogonal surfaces are measured by gap sensors. It is then rotated by 90° and measured. After two additional rotations (by 90° each in the same direction) and repeated sensor measurements, all of the surfaces of the polygon are in the end measured twice. At each rotation, there exists a geometrical relationship between an inner angle of the polygon and the two measured angles, i.e. the γ’s (equation (3)). A total of four relationships (for four inner angles) and the 2π sum of the four angles give a simple final equation, as expressed by equation (4). αxy is the out-of-squareness angle and β denotes the inner angle of the polygon.
αxy = βBA + γ1 + γ2 αxy = βAD + γ3 + γ4 αxy = βDC + γ5 + γ6 αxy = βCB + γ7 + γ8 αxy =
1 4
8 ⎛ ⎞ ⎜⎜2π + ∑ γk ⎟⎟ ⎝ k=1 ⎠
(3)
(4)
3. System description The stage to be calibrated is a 3-DOF, linear motor-driven, airbearing guided stage. The key specifications of this stage are that it has following errors less than 50 nm (velocity ripple less than 0.1%), straightness errors less than 1 μm and an ortho gonality error of less than 1 arcsec for x = 400 mm (step) and y = 700 mm (scan) strokes. To achieve this level of precision, 1000 mm (y) and 550 mm (x) bar mirrors were utilized and real-time Δx and Δy compensation using laser feedback control was applied. SPiiPlus [11] was used as a motion controller. The wavelength variation of the laser and the beam instability were kept at a minimum through strict environmental control (23 ± 0.1 °C, 50% ± 1%
2. Self-calibration method 2.1. Self-calibration of straightness
Figure 1 shows the concept of the self-calibration of the straightness of a bar mirror. A gap (capacitive) sensor is attached to the moving stage and used to measure
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Figure 2. The self-calibration of orthogonality.
Figure 3. Yaw error caused by angle between Y1 and Y2 interferometer head.
RH). The stage’s scan performance, i.e. following error and yawing, was controlled by dual laser interferometers (Y1 and Y2), while its y-directional straightness (the straightness error that occurs when the stage is driven in the y-direction) was controlled by one x interferometer (see figure 2 for the configuration). The laser interferometer system is composed of Renishaw RLE10 and its environmental compensation unit, RCU10 [12]. Glass scales are also used for homing and to provide extra positional information.
Figure 4. Y-direction yaw calibration using a laser calibration system.
Measurements were made every 70 mm out of the 700 mm y stroke. The measurement was made three times (for both forward and backward directions) and showed that the y-directional yaw error was 0.8 arcsec on average with a repeatability of 0.2 arcsec. Because the y-directional yaw for infinitesimally small steps was controlled, the 0.8 arcsec yaw was assumed to be an equivalent low-frequency turning of the stage rather than high-frequency yaw under control. This turning was suspected to be a result of the cosine error of the interferometers. From figure 3 and equation (5), the misaligned angle (θ) during the installation of the Y1 and Y2 interferometers causes yaw motion (Φ) when the stage moves a distance L. Hence, the installation error (θ) is estimated to be 250 arcsec. To compensate for this, a scale factor of 0.999 999 2662 is multiplied at the Y2 output in the controller. With these new factors, the y-directional yaw was reduced to its level of repeatability, i.e. 0.2 arcsec (figure 4).
4. Straightness calibration 4.1. Yaw error elimination
Before compensating the x- and y-directional straightness errors of the stage using a self-calibration method, the x- and y-directional yaw errors (the yaw errors that occur when the stage is driven in the x- and y-directions, respectively) were eliminated such that they did not affect the calibration. First, the y-directional yaw was measured using a laser calibrator (Renishaw XL-80 [12]). The x position of the stage was fixed at the center position of the full x stroke. 3
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Figure 5. The self-calibration experiment setup for (a) the y- and (b) the x-directional straightness (capacitive sensor is not shown).
Figure 6. Gap sensor measurement data: (a) before and (b) after the reversal of the straightedge. 4
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Figure 7. The result of the y-directional straightness.
Laser feedback control for positional accuracy, yaw and straightness was utilized. The scan speed of the stage was kept under 10 mm s−1 with its control gain adjusted for faster settlement and better in-position stability at each measurement step, ensuring the seamless capture of the signals. With regard to the gap sensor assembly, it was prevented from colliding with the straightedge throughout the full strokes of the stage. The position of the gap sensor was finetuned such that its voltage output could vary at around 5 V for the full x and y test strokes. The measurement height of the gap sensor was also adjusted to the middle of the effective width of the edge surface so that it could measure the same profile before and after the reversal.
The x-directional yaw was then measured. An average of 2.0 arcsec was measured at the center position of the full y stroke. It is larger than the y-directional yaw because it is augmented by the x-bar mirror deflection, which creates Y1 and Y2 feedback to generate the yaw motion of the stage as the stage moves along the x direction. To compensate for this x-directional yaw, the Y1–Y2 offsets required along the x steps were added to the Y2 interferometer feedback data. As a result, the x-directional yaw was reduced to 0.2 arcsec.
⎛ L (1 − cos θ ) ⎞ ⎟ φ = sin−1 ⎜ ⎝ ⎠ l
(5)
With the x- and y-directional yaw errors made almost equal to their levels of repeatability, self-calibration could be applied.
4.3. Results 4.3.1. Y-directional straightness. Figure 6 shows the mea-
surement results before (a) and after (b) the reversal of the straightedge. The results show that the repeatability of the tests was very good. Average values found were inserted in equation (2) for the straightness error calculations of the bar mirror and the straightedge. Slopes caused by non-parallelism between the bar mirror and the straightedge were subtracted using first-order regression functions (shown in the plots). Figure 7 shows the final calibration result of the y-directional straightness. The circle-dotted line represents the straightness of the y-bar mirror while the star-dotted line represents the straightness of the straightedge. The result demonstrates that the straightness of the bar mirror is about 2.2 μm and that of the straightedge is about 3.2 μm.
4.2. Setup and procedure
Figure 5 shows images of the straightedge being installed for the self-calibration of (a) the y- and (b) the x-directional straightness. The figure also shows the bar mirrors and some of the installation jigs, but not the gap sensor and its adapter. As discussed earlier, the goal of this procedure is to measure the varying gap created by the straightness error of the stage and that of the straightedge. Measurements were repeated 15 times for each side of the straightedge. The positional data of the stage and the gap sensor data were recorded every 10 mm step. Sufficient time was allowed for the temperature to settle after each hardware setup. The x and y home positions were set to constant offsets from the positions where the mechanical limits were located. In this way, the reference positions can be maintained throughout the calibration process.
4.3.2. X-directional straightness. Figure 8 shows the mea-
surement results before (a) and after (b) the reversal of the 5
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Figure 8. Gap sensor measurement data: (a) before and (b) after the reversal of the straightedge.
straightedge. Again, each result demonstrates very nice repeatability. Average values were found and inserted into equation (2) for the straightness error calculations of the bar mirror and straightedge. Slopes generated by non-parallelism between these two surfaces were subtracted. Figure 9 shows the final result of the x-directional straightness. The circle-dotted line represents the straightness of the x-bar mirror while the star-dotted line represents that of the straightedge. The result demonstrates that the straightness of the bar mirror is about 4.5 μm and that of the straightedge is about 4.2 μm. If the same edge surface of the straightedge was used for both the x- and y-directional straightness calibrations,
these two straightness results could have been comparable. However, different surfaces were used in the two tests. 5. Orthogonality calibration 5.1. Setup and procedure
Application of the self-calibration method to orthogonality calibration requires a total of four sets of measurements. In each measurement set, two perpendicular edges are measured 10 times (figure 10). The effective size of the polygon was 230 × 190 mm2 and thus covered only part of the full stage strokes. However, it 6
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Figure 9. The result of the x-directional straightness.
Figure 10. Orthogonality calibration (polygon is 90° rotated in the right image).
γ8 (the measurement of edge C after rotation of the polygon by 270°). This was due to unusual vibration of the stage which occurred when measuring those parameters. The test results could have been improved with additional tests, but the results were kept as the slopes of the trend lines would have remained the same even with the improved results. Using average values for all γ’s (listed in table 1), the outof-squareness angle, α, was obtained from equation (6). It was found to be an obtuse angle.
was assumed to be large enough to represent the full range out-of-squareness of the stage. This assumption was adequate because the results showed linear relationships between the sensor values and the measurement steps (figure 11). Gap sensors of a longer measurement range (200 μm instead of 50 μm) were used to avoid collisions. The sensor gap was adjusted such that its voltage output changed at around 5 V. Other conditions (e.g. laser feedback control, scan speed, measurement step, environment control, reference position) were set similarly to those applied for the straightness calibration. Figure 10 shows the polygon, gap sensor and the jigs installed for the measurement procedure.
6. Evaluation Compensation data to be embedded in the controller were created from the calibration results. The x- and y-directional straightness data were directly added to the program. The orthogonality result was transformed to a first-order linear function and added to the y-directional straightness data.
5.2. Result
Figure 11 shows all the measurement results. The repeatability was in general very good except for the case of γ6 (the measurement of edge D after rotation of the polygon by 180°) and 7
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(a)
(b)
(c)
(d) Figure 11. Results of orthogonality calibration. Table 1. Angle (γk) between the reference plane of the bar mirror and a plane of the polygon.
γ1 γ2 γ3 γ4 γ5 γ6 γ7 γ8
0.000 8761 −0.000 1628 −0.000 4670 0.001 1728 0.000 7378 −0.000 0192 0.000 7369 −0.000 0241
Figure 12. 60 × 60 μm cross mark (each one is located at four
corners of the patterned glass).
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Figure 13. The y-directional straightness evaluated by a calibrator.
Figure 14. The x-directional straightness evaluated by a calibrator.
Figure 15. The y-directional straightness evaluated by TFT glass.
The results of the straightness compensations were evaluated using a laser calibrator. The result of the orthogonality compensation was evaluated using a 300 × 400 mm2 TFTpatterned LCD glass by probing four corners of an outer rectangular pattern with a 20×, 220 nm resolution microscope (the type is shown in figure 12). The y-directional straightness was confirmed with this TFT glass.
the evaluation result of the x-directional straightness. The measurement was run three times (for both forward and backward directions) for the range of 380 mm and was found to be 0.4 μm. In both cases, the repeatability was measured in the range of 0.4~0.5 μm. Both results satisfied the specifications. The measurement accuracy of the calibrator was ±0.5 ppm [12]. 6.2. Evaluation using a TFT-patterned glass
6.1. Evaluation using laser calibrator
The evaluation result of the orthogonality using the TFTpatterned glass was found to be 2.48 arcsec. One assumption here is that the reference TFT glass is perfectly orthogonal. Further manual compensation was done to meet the level of
Figure 13 shows the evaluation result of the y-directional straightness. The measurement was run three times for the range of 630 mm and was found to be 0.8 μm. Figure 14 shows 9
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References
orthogonality of the TFT glass. As a result, the orthogonality error was reduced to 0.35 arcsec, satisfying its specifications. Figure 15 shows the evaluation result of the y-directional straightness error which was confirmed with the TFT glass. The amount of deviation of the line pattern from a fixed point in the FOV was identified via image processing. The result (0.7 μm) was in good agreement with that measured with the calibrator (0.8 μm).
[1] Slocum A H 1992 Precision Machine Design (Englewood Cliffs, NJ: Prentice-Hall) [2] Evans C J and Hocken R J 1996 Self-calibration: reversal, redundancy, error separation and absolute testing Ann. CIRP 45 617–34 [3] Hocken R J and Borchardt B R 1979 On characterizing measuring machine geometry NBSIR 79–1752 (Washington, DC: National Bureau of Standards) [4] Lieberman B 1978 Quality assurance procedures for MEBES J. Vac. Sci. Technol. 15 913–6 [5] Raugh M R 1984 Absolute 2D sub-mircron metrology for electron beam lithography Proc. SPIE 480 145–62 [6] Raugh M R 1985 Absolute 2D sub-micron metrology for electron beam lithography: a calibration theory with application Precis. Eng. 7 3–13 [7] Takac M T 1993 Self-calibration in 1D Proc. SPIE 2087 80–6 [8] Ye J, Takac M and Berglund C N 1997 An exact algorithm for self-calibration of 2D precision metrology stages Precis. Eng. 20 16–32 [9] Ruijl T 2001 Ultra precision coordinate measuring machine PhD Thesis TU Delft pp 107–12 [10] Hume K J 1974 Metrology With Autocollimators (London: Hilger and Watts) [11] www.acsmotioncontrol.com/, ACS Motion Control home page (2008) [12] www.renishaw.com/, Renishaw home page (2008)
7. Conclusion A self-calibration method was successfully applied to compensate for the straightness and orthogonality errors of a three-axis laser feedback stage system having rectangular strokes 400 mm and 700 mm. In particular, x- and y-directional yaw errors were carefully eliminated before the application of the method. Calibration results showed that the y and x bar mirror straightness and orthogonality errors were 2.2 μm, 4.5 μm and 0.040 8314°, respectively. After compensation, the corresponding results became 0.8 μm (y-dir.), 0.5 μm (x-dir.) and 2.48 arcsec. Compensations for these errors were conducted and the final results were evaluated using a laser calibrator and a TFT-patterned LCD glass. The results were found to fully satisfy the specifications.
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