Reduction in Cross-Talk Errors in a Six-Degree-of-Freedom Surface Encoder
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Abstract
This paper presents the reduction in cross-talk errors in the angular outputs existing in the previously designed six-degree-of-freedom (DOF) surface encoder for nanopositioning and nanometrology. The six-DOF surface encoder is composed of a planar scale grating and an optical sensor head with a reference grating, a displacement assembly and an angle assembly. The diffracted beams from both the scale and the reference gratings are received by the displacement assembly for measurement of the primary XYZ translational motions. The angle assembly only receives the diffracted beams from the scale grating for measurement of the secondary θ_{X}θ_{Y}θ_{Z} angular motions. In this paper, at first, the cross-talk errors in the angular measurement results of the previous surface encoder are identified to be caused by the diffracted beams from the reference grating leaking into the angle assembly due to the imperfection of the polarization components of the sensor head. An improved design of the sensor head is then carried out to reduce the cross-talk errors by changing the position of the angle assembly in the sensor head. The sensor head is further optimized by replacing the beam splitter located in front of the angle assembly from a cube type to a plate type. Experimental results have demonstrated that the cross-talk errors were reduced from 3.2 arc-seconds to 0.02 arc-second.
Keywords
Six-DOF surface encoder Cross-talk errors1 Introduction
Ultra-precision products such as optical instruments and semiconductor devices are composed of parts with a variety of geometries from fundamental flat and/or spherical shapes to the complex free-form surfaces [1, 2, 3]. Many of such parts are required to be manufactured in a high precision up to sub-micrometer or even nanometer. The related nanomanufacturing technologies have been developed in recent years [4, 5]. On the other hand, due to the tight tolerances in nanomanufacturing, nanometrology of the manufactured parts is of high priority for quality control of the parts as well as for process control of nanomanufacturing [6]. Taking into consideration the complex three-dimensional (3D) shapes of the parts, nanomanufacturing machines are required to generate the corresponding 3D tool paths by synchronizing the multi-axis motions of the machine axes, i.e., to carry out multi-axis and multi-degree-of-freedom (multi-DOF) precision positioning of the tool with respect to the workpiece. Nanopositioning and the related nanometrology are therefore necessary for this purpose [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
The three-axis XYZ linear motions by linear slides/stages are the most fundamental motions in a nanomanufacturing machine although rotary motions by spindles/swivel tables are also standard motions for multi-axis machine tools. In a three-axis XYZ linear positioning system, θ_{X}, θ_{Y} and θ_{Z} angular motions are associated with the primary X, Y and Z translational motions. For high-precision positioning, such a system must be operated with full closed-loop control of not only the three-DOF XYZ translational motions (ΔX, ΔY and ΔZ) but also the three-DOF θ_{X}θ_{Y}θ_{Z} angular motions (Δθ_{X}, Δθ_{Y} and Δθ_{Z}) [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. A six-DOF measurement system is thus essential for this purpose. A conventional multi-DOF measurement system is generally composed of multiple single-axis sensors such as laser interferometers and autocollimators [28]. However, the usage of multiple sensors leads to an increase in the size, complexity and cost of the measurement system as well as those of the overall positioning system. It is also difficult to avoid the Abbe error in such a measurement system because it is difficult for all the sensors to measure the same position [7]. The influence of environmental fluctuations on laser interferometers and autocollimators due to their long and changing working distances is another shortcoming of the conventional measurement system.
In order to solve these problems, an optical sensor referred to as the six-DOF surface encoder [27] for XYZ three-axis precision positioning had previously been proposed by the authors. With a two-dimensional (2D) scale grating and a reference grating manufactured by FTS-based diamond turning [31] or optical interference lithography [32, 33], the surface encoder can carry out the six-DOF measurement by using a single sensor head with a short and constant working distance. The optical system of the sensor consists of a displacement assembly and an angle assembly. In the displacement assembly, the diffracted beams from the scale grating, which are referred to as the scale beams, interfere with those from the reference grating, which are referred to as the reference beams, for measurement of the primary three-axis (X, Y and Z) translational displacements [34, 35]. In the angle assembly, the diffracted beams from the scale grating are employed for measurement of the secondary three-axis angular (θ_{X-}, θ_{Y-}, and θ_{Z-}) displacements based on the principle of three-axis laser autocollimation [36].
In the previous research, the six-DOF surface encoder system had been designed in a compact size for integrating in a surface motor-driven planar stage [27]. With this initial design, the surface encoder had achieved a translational displacement resolution better than 2 nm and an angular displacement resolution of approximately 0.1 arc-second, which had satisfied the design requirements on resolutions. On the other hand, however, variations had been detected in the outputs of the angle assembly when a translational displacement had been applied to the scale grating with respect to the sensor head. This indicates the existence of the cross-talk error components of the surface encoder. The amplitudes of the cross-talk error components, which had been up to 3.2 arc-seconds, are quite large compared with the angular resolutions of the surface encoder and should be reduced. As the goal for the cross-talk error reduction, it is aimed to reduce the amplitude of the cross-talk error component to be smaller than 0.1 arc-second so that it can be consistent with the angular resolution level. Since the cross-talk error was the largest uncertainty source for the angular outputs of the surface encoder, the achievement of such a goal is also important for assurance of the accuracy of the surface encoder, which is designed to be better than 1 arc-second based on the requirement from the surface motor-driven planar stage.
In this paper, the design of the optical layout of the sensor head of the surface encoder is improved by changing the position of the angle assembly in the sensor head for reduction in the cross-talk error components in the angular outputs of the surface encoder based on an analysis of the reasons for causing the cross-talk error components in the initial design. The improved design is then optimized to further reduce the remained cross-talk error components by replacing a cube-type beam splitter with a plate-type beam splitter. Experimental results are presented for demonstrating the results of cross-talk error reduction in both the improved and the optimized designs.
2 Analysis of Cross-Talk Errors in the Previous Design of Six-DOF Surface Encoder
Based on the experimental results shown above, the reason for the periodical cross-talk errors is analyzed. In the ideal situation, all the reference beams will be blocked by Polarizer1 as shown in Fig. 2.
Similar light spots can be observed on QPD2 for detecting θ_{Z}. In the ideal situation without leakage beam, only the zeroth-order reference beam is received by the angle assembly. The beam is focused to be a small light spot on the QPD by the collimator objective, which is referred as the scale beam spot in the figure. The centroid point C_{i} of the light spot will be the center of the spot. Based on the principle of laser autocollimation, a θ_{X} motion of the scale grating will cause a linear displacement of the centroid on the QPD along the V-direction. Similarly, a θ_{Y} motion will cause a linear displacement of the centroid along the W-direction. The 2D displacements of the centroid are detected by the QPD, from which the θ_{X} and θ_{Y} can be evaluated.
Since a blu-ray laser diode with a λ of 405 nm was employed in the sensor head, \(P_{{Z - \theta_{X} }}\) and \(P_{{Z - \theta_{Y} }}\) are thus calculated to be 202.5 nm. These are in a good correspondence with the values of P_{θX} and P_{θY} in Fig. 4.
For λ of 405 nm and g of 0.57 μm, P_{θZ} is calculated to be approximately 238 nm. This value also well agrees with that of \(P_{{Z - \theta_{Z} }}\) and Fig. 4. On the other hand, it is observed in Fig. 4 that the amplitude of the periodic cross-talk error component in θ_{Z} is larger than those in θ_{X} and θ_{Y}. This can be caused by the difference in the diffraction efficiencies of the zeroth-order and the first-order diffracted beams. The alignment errors of the optical components in the sensor head are other possible reason.
The calculated value of \(P_{{XY - \theta_{Z} }}\) (0.57 μm) is in a good correspondence with the experimental result shown in Figs. 5 and 6.
Consequently, the cross-talk errors in the outputs have been successfully identified to be caused by the interference between the scale beams and the leakage reference beams to the angle assembly based on the above analysis. To reduce the cross-talk errors, it is thus necessary to completely prevent the entry of the reference beams into the angle assembly. It should be noted that the above analysis is carried out for identification of the reasons for the cross-talk errors, based on which the improvement in the surface encoder shown in the next section can be carried out. For this reason, to identify the periods of the intensities of the interference waves was the highest priority in the analysis. It would be an interesting future work to make a more accurate and quantitative analysis through accurately determining the leakage rate k as well as the intensities of the diffracted beams experimentally so that not only the periods but also the amplitudes of the intensities of the interference waves can be identified.
3 Reduction in Cross-Talk Errors in an Improved Six-DOF Surface Encoder
An improved design of the sensor head is proposed in this paper to prevent the entry of the reference beams into the angle assembly based on the fact that the imperfection of the polarization components (the PBS, QWPs, polarizer) is unavoidable, which had been the reason for causing the leakage of the reference beams in the previous sensor head.
It is known that a certain percentage (typically 4%) of the light incident normally on an air–glass interface will be reflected back [37]. This will happen at the cube surfaceT of BS2. Therefore, a part of Beam2, which is denoted as Beam2T, will enter the angle assembly. In a practical case, Beam2T can be received by both QPD C for measuring the θ_{X} and θ_{Y} motions and QPD D (or QPD E) for measuring the θ_{Z} motion. Beam2T will then interference with the scale zeroth-order and first-order beams to generate interference fringes on QPD C and QPD D (or QPD E).
When ΔZ is applied to the scale grating, the centroid point on QPD C will oscillate with a period of λ/2 as shown in Eq. (28), which will be the period for the periodic cross-talk error components in Δθ_{X}, Δθ_{Y}. On the other hand, when ΔZ is applied to the scale grating, the centroid point on QPD D (or QPD E) will oscillate with a period of λ/(1 + cosθ) as shown in Eq. (29). This will be the period for the cross-talk errors in Δθ_{Z}. The analysis results are well consistent with the results shown in Figs. 16 and 17. It should be noted that the differences between m_{0}, n_{0} and m_{1}, n_{1} will make a difference between the amplitude of the periodic cross-talk error component in Δθ_{X}, Δθ_{Y} and that in Δθ_{Z}, which are shown in Figs. 16 and 17. Z_{S}, Z_{R} and F_{S} were 0.25 nm, 2.49 μm and 0.40 μm^{-1}, respectively. Based on the spatial wavelengths of the dominant periodic components in Fig. 17, P_{Z−θX}, P_{Z−θY} and P_{Z−θZ} were evaluated to be around 200 nm. The residual periodic cross-talk error components caused by ΔX and ΔY of the scale grating were 0.074 arc-second and 0.046 arc-second, respectively, which achieved the targeted 0.1-arc-second goal. However, the residual periodic cross-talk error component caused by ΔZ of the scale grating was identified to be 0.38 arc-second, which was larger than the targeted goal. Therefore, an optimized design shown in the next section is made to reduce this error.
4 An Optimized Six-DOF Surface Encoder with Minimized Cross-Talk Errors
Cross-talk errors with various optical designs of six-DOF surface encoder for ΔZ
θ_{X} arc-second | θ_{Y} arc-second | θ_{Z} arc-second | |
---|---|---|---|
The previous design | 0.38 | 0.48 | 3.24 |
Improved design | 0.074 | 0.046 | 0.38 |
Optimized design | 0.0070 | 0.0074 | 0.0196 |
5 Conclusions
In this paper, reduction in cross-talk errors in a six-DOF surface encoder for a planar motion stage has been carried out. The cross-talk errors in the outputs of previous prototype of six-DOF surface encoder have been successfully identified based on theoretical analysis. The cross-talk errors have been caused by the interference between the scale beams and the leakage reference beams to the angle assembly. In order to reduce the cross-talk errors, an improved six-DOF surface encoder sensor head has been proposed with preventing the entry of the reference beams into the angle assembly.
In the optimized design of the sensor head, cross-talk errors have been still remained. The interference between the scale light and the internal reflection of the laser beam in a cube-type beam splitter has been identified to be the cause of the cross-talk errors. The cube-type beam splitter in the sensor head has been replaced with a plate-type beam splitter for avoiding the influence of the internal reflection. As a result, the cross-talk errors have been successfully reduced. The resultant cross-talk errors with the optimized six-DOF surface encoder have been 0.0070 arc-second, 0.0074 arc-second and 0.0196 arc-second for θ_{X}, θ_{Y} and θ_{Z}, respectively. The experimental results have demonstrated that the targeted goal of 0.1 arc-second has been achieved in all the three directions.
Notes
Acknowledgements
This research is supported by Japan Society for the Promotion of Sciences (JSPS) KAKENHI.
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