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Reduction in Cross-Talk Errors in a Six-Degree-of-Freedom Surface Encoder

  • Hiraku MatsukumaEmail author
  • Ryo Ishizuka
  • Masaya Furuta
  • Xinghui Li
  • Yuki Shimizu
  • Wei Gao
Original Articles
  • 52 Downloads

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 θXθYθZ 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 errors 

1 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, θX, θY and θZ 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 θXθYθZ angular motions (ΔθX, ΔθY and ΔθZ) [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 (θX-, θY-, and θZ-) 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

Figure 1 shows the basic concept of previously designed surface encoder for measuring the six-DOF motions of a moving scale grating [27]. The light from a laser diode (LD) is divided into two parts at point C on the hypotenuse surface of a cube beam splitter (BS1). The divided light beams are projected onto the scale and the reference gratings, respectively. Since both the reference and scale gratings are planar gratings having identical grating structures along the X- and Y-directions, diffracted beams are generated by the scale grating and the reference grating in the XZ- and YZ-planes, respectively. Only the zeroth-order and the positive and negative first-order diffracted beams in the XZ-plane are shown in the figure for the sake of clarity. As shown in the figure, the positive and negative first-order diffracted beams in the XZ-plane from the scale grating recombine with those from the reference grating at points A and B, respectively. The combined beams are then interference with each other to generate two sets of interference signals. The intensities of the two interference signals are detected by two single-cell photodiodes (PD1, PD2) in the displacement assembly after being reflected by another cube beam splitter (BS2). Since the phase of each of the interference signals is a function of the translational motions ΔX and ΔZ, ΔX and ΔZ can then be solved from the outputs of PD1 and PD2 [32]. Similarly, ΔY and ΔZ can be obtained from the interference signals generated by the positive and negative first-order diffracted beams in the YZ-plane. Consequently, the three-DOF translational motions ΔX, ΔY and ΔZ can be simultaneously measured. For measurement of the three-DOF angular motions ΔθX, ΔθY, ΔθZ, the scale zeroth- and positive first-order diffracted beams are received by the angle assembly for simultaneously detecting the angular motions based on the principle of three-axis laser autocollimation [28]. The zeroth-order diffracted beam is focused onto quadrant photodetector 1 (QPD1) for detection of ΔθX, ΔθY. The positive first-order diffracted beam is focused onto quadrant photodetector 2 (QPD2) for detection of ΔθZ.
Fig. 1

Basic concept of the previous design for the six-DOF surface encoder

Since only the scale diffracted beams are necessary in the angle assembly for the angular motion measurement, optical isolation had been adopted in the previous design by using polarization components to prevent the entry of the reference diffracted beams into the angle assembly [27]. As shown in Fig. 2, a cube polarizing beam splitter (PBS) is employed to replace BS1 in Fig. 1 for dividing the light from the LD into a P-polarized beam and an S-polarized beam. The divided beams are projected onto the scale and reference gratings after passing through quarter waveplates QWP1 and QWP2, respectively. The diffracted beams reflected back from the scale grating are changed to S-polarization by QWP1 so that the scale beams can be reflected at the hypotenuse surface of the PBS toward BS2. The reference beams, which are changed to P-polarization by QWP2, transmit through the PBS toward BS2. Half of the combined scale and reference beams are reflected at the BS and then received by the displacement assembly. The other half of the combined scale and reference beams pass through BS2 toward the angle assembly. Polarizer1 is added in front of the angle assembly to block the reference beams. The transmission axis of Polarizer1 is adjusted to be parallel with the polarization direction of the scale beams and orthogonal to that of the reference beams so that only the scale beams can be received by the angle assembly.
Fig. 2

Optical isolation in the previous design for preventing the entry of the reference diffracted beams into the angle assembly

However, cross-talk errors had been confirmed to exist in the outputs of the angular assembly of the previous surface encoder with the design shown in Fig. 2 [27]. Here, the data are analyzed to identify the cause of the cross-talk errors. Figure 3 shows the measured outputs of the angular assembly of the previous surface encoder when only a Z-direction translational displacement was applied to the scale grating by a PZT stage while the reading head was kept stationary. The angular error motions of the PZT stage were confirmed to be sufficiently small. The output signals were smoothed by the simple moving average method. The number of averaged points was 100. Periodical cross-talk error components can be observed in the θX-, θY- and θZ-outputs. In the ideal case, the angular outputs are expected to be constant, i.e., the output curves should be straight lines parallel to the horizontal axis of the figure when an X-, Y- or Z-direction translational displacement is applied to the scale grating. In the actual case, however, deviations of the angular outputs will be associated with the translational displacement, which are error components of the angular outputs. The deviation of each angular output is defined as the cross-talk error. As shown in Fig. 3, cross-talk error components can be observed in the θX-, θY- and θZ-output curves. The peak-to-valley (PV) amplitude of the deviation of each angular output curve, i.e., the cross-talk error curve, in Fig. 3 was calculated to make a quantitative evaluation of the error. The cross-talk errors were thus evaluated to be 0.38 arc-second, 0.48 arc-second and 3.24 arc-second in the θX-, θY- and θZ-outputs, respectively.
Fig. 3

Measured cross-talk errors in the outputs of the angle assembly of the previous surface encoder when only a Z-direction translational displacement was applied to the scale grating

As shown in Fig. 3, the cross-talk error curves are dominated by periodic components with specific periods PZθX, PZθY and PZθZ along the Z-axis. Since each cross-talk error curve is regarded as a function of the Z-displacement, it is effective to carry out discrete Fourier transform (DFT) of the cross-talk curve in the spatial frequency domain for evaluation of the period. By using DFT, the cross-talk error in Fig. 3, as a function of Z-displacement, is transformed into a function of a series of discrete spatial frequencies with an equal spatial frequency spacing FS having a unit of 1 μm-1. Each of the cross-talk error curves in Fig. 3 has been acquired with an equal sampling interval of ZS over a range of ZR along the Z-axis. FS is then equal to 1/ZR. The maximum spatial frequency and the minimum spatial wavelength of the spatial frequency components are 1/2ZS and 2ZS, respectively, where the special wavelength with a unit of μm is a reciprocal of the spatial frequency. Figure 4 shows the results of the DFT in Fig. 3. ZS, ZR and FS were 6.7 nm, 1.0 μm and 1.0 μm-1, respectively. The horizontal axis shows the spatial wavelengths of the spatial frequency components, and the vertical axis shows the corresponding amplitudes. Based on the spatial wavelengths of the dominant periodic components in Fig. 4, PZθX, PZθY and PZθZ were evaluated to be approximately 200 nm, 200 nm and 250 nm, respectively.
Fig. 4

DFT of the cross-talk errors in Fig. 3. aθX, bθY, cθZ

The results when only an X-direction translational displacement was applied to the scale grating by a PZT stage are shown in Fig. 5. In this case, the periodic cross-talk error is only observed in θZ and the PV amplitude was 3.12 arc-seconds. The DFT of θZ is shown in Fig. 6. Each of the cross-talk error curves in Fig. 5 has been acquired with an equal sampling interval of XS over a range of XR along the X-axis. The spatial frequency spacing FS is then equal to 1/XR. Figure 6 shows the results of the DFT in Fig. 5. XS, XR and FS were 6.7 nm, 1.0 μm and 1.0 μm-1, respectively. Based on the spatial wavelengths of the dominant periodic component in Fig. 6, PXθZ was evaluated to be approximately 500 nm.
Fig. 5

Measured cross-talk errors in the outputs of the angle assembly of the previous surface encoder when only an X-direction translational displacement was applied to the scale grating

Fig. 6

DFT of the cross-talk errors θZ in Fig. 5

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.

However, in a practical situation shown in Fig. 7, due to the imperfection of the PBS and QWP2, the zeroth-order and the first-order reference beams arriving at Polarizer1 will include not only the expected P-polarization component but also the unexpected S-polarization component. The S-polarization component will pass through the polarizer and enter the angle assembly, which is referred to the leakage beam in the figure.
Fig. 7

Beam leakage at the angle assembly

Figure 8 shows the light spots on QPD1 for detecting θX and θY.
Fig. 8

The light spot on QPD under the ideal situation without leakage beam

Similar light spots can be observed on QPD2 for detecting θZ. 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 Ci of the light spot will be the center of the spot. Based on the principle of laser autocollimation, a θX motion of the scale grating will cause a linear displacement of the centroid on the QPD along the V-direction. Similarly, a θY 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 θX and θY can be evaluated.

On the other hand, in a practical situation with the leakage beam, an additional light spot will be formed on the QPD, which is referred to as the reference leakage beam spot in Fig. 9. Since the two light spots have the same polarization direction, interference fringes will be generated on the QPD due to the interference of the two light spots. The number of fringes is determined by the alignment of the optical components in the sensor head. If the optical components are perfectly aligned, the reference beam and the scale beam will be parallel and completely overlap with each other, resulting in a zero interference fringe. In an actual case, there will be a certain number of fringes due to the imperfect alignment. In either case, the centroid point is now the weighted center of the scale beam spot, the reference leakage beam spot and the interference fringes, which is denoted as Cp in the figure. Similar to Ci in Fig. 8, Cp will move on the QPD in responding to the θX and θY motions of the scale grating. However, additional displacements of Cp will also be caused by the variations of the interference fringes, i.e., the variations of the phase in the interference signal of the zeroth-order scale beam and the zeroth-order leakage reference beam.
Fig. 9

The light spots on QPD under the actual situation with leakage beam

Here, we assume that the wavefronts of the reflected zeroth-order and positive first-order diffracted beams from the reference grating are denoted by Er0 and Er1, respectively. Without considering the initial phases, Er0 and Er1 can be expressed as
$$E_{{{\text{r}}0}} = E_{0}$$
(1)
$$E_{{{\text{r}}1}} = E_{0}$$
(2)
where E0 is the amplitude of the wave. The diffraction efficiencies for the zeroth-order beam and the first-order beam are assumed to be equal with each other. Since the reference grating is kept stationary, Er0 and Er1 will not change when the scale grating is moved.
Let the wavefronts of the reflected zeroth-order and positive first-order diffracted beams from the scale grating at the initial position of the scale grating to be denoted by Ei−s0 and Ei−s1, respectively. Without considering the initial phases at this position, Ei−s0 and Ei−s1 can be expressed as
$$E_{{i - {\text{s}}0}} = E_{0}$$
(3)
$$E_{{i - {\text{s}}1}} = E_{0}$$
(4)
When the scale grating is moved with a translational motion ΔZ along the Z-directions, optical path differences shown in Fig. 10 will be generated, resulting in the following phase shifts ϕZ−s0 in the zeroth-order diffracted beam and ϕZ−s1 in the positive first-order diffracted beams:
Fig. 10

Phase shifts caused by ΔZ of the scale grating. a The scale zeroth-order beam. b The scale positive first-order beam

$$\phi_{{Z - {\text{s}}0}} = \frac{2\pi }{\lambda }2\Delta Z$$
(5)
$$\phi_{{Z - {\text{s}}1}} = \frac{2\pi }{\lambda }\left( {1 + \cos \theta } \right)\Delta Z$$
(6)
where λ is the wavelength of the LD and θ is the diffraction angle expressed as follows:
$$\theta = \arcsin \frac{\lambda }{g}$$
(7)
where g is the pitch of the grating.
If the scale grating is moved with translational motions ΔX and ΔY along the X- and Y-directions, the optical path differences shown in Fig. 11Y is not shown in the figure for clarity) will generate the following phase shifts in the scale zeroth-order and the positive first-order diffracted beams:
Fig. 11

Phase shifts caused by ΔX and ΔY of the scale grating. a The scale zeroth-order beam. b The scale positive first-order beam

$$\phi_{{XY - {\text{s}}0}} = 0$$
(8)
$$\phi_{{XY - {\text{s}}1}} = \frac{2\pi }{g}\left( {\Delta X + \Delta Y} \right)$$
(9)
The wavefront functions of the scale zeroth-order and positive first-order diffracted beams can then be written as:
$$E_{{{\text{s}}0}} = E_{0} {\text{e}}^{{i\left( {\phi_{{z - {\text{s}}0}} + \phi_{{xy - {\text{s}}0}} } \right)}} = E_{0} {\text{e}}^{{i\frac{2\pi }{\lambda }2\Delta Z}}$$
(10)
$$E_{{{\text{s}}1}} = E_{0} {\text{e}}^{{i\left( {\phi_{{z - {\text{s}}1}} + \phi_{{xy - {\text{s}}1}} } \right)}} = E_{0} {\text{e}}^{{i\frac{2\pi }{g}\left( {\Delta X + \Delta Y} \right)}} {\text{e}}^{{i\frac{2\pi }{\lambda }\left( {1 + \cos \theta } \right)\Delta Z}}$$
(11)
Assuming that the zeroth-order and the positive first-order reference beams have the same leakage rate of k, the leakage reference beams can thus be expressed by
$$E_{{{\text{l}}0}} = kE_{0}$$
(12)
$$E_{{{\text{l}}1}} = kE_{0}$$
(13)
The wavefront function of a point of the interference fringes generated on QPD1 by El0 and El0 can be written as:
$$E_{{{\text{ls}}0}} = E_{{{\text{l}}0}} + E_{{{\text{s}}0}}$$
(14)
Similarly, the wavefront function of a point of the interference fringes generated QPD2 by El+1 and Es+1 can be written as:
$$E_{\text{ls1}} = E_{{{\text{l}}1}} + E_{\text{s1}}$$
(15)
The intensities of the interference waves on QPD1 (Ils0) and QPD2 (Ils+1) can thus be obtained as:
$$\begin{aligned} I_{{{\text{ls}}0}} & = E_{{{\text{ls}}0}} \cdot \overline{{E_{{{\text{ls}}0}} }} \\ & = E_{0}^{2} \left( {1 + k^{2} + 2k\cos \frac{2\pi }{\lambda }2\Delta Z} \right) \\ \end{aligned}$$
(16)
$$\begin{aligned} I_{{{\text{ls}}0}} & = E_{{{\text{ls}}1}} \cdot \overline{{E_{{{\text{ls}}1}} }} \\ & = E_{0}^{2} \left[ {1 + k^{2} + 2k\cos \left( {\frac{2\pi }{g}\left( {\Delta X + \Delta Y} \right) + \frac{2\pi }{\lambda }\left( {1 + \cos \theta } \right)\Delta Z} \right)} \right] \\ \end{aligned}$$
(17)
When a Z-direction translational motion ΔZ, which is referred to as an out-of-plane motion, is applied to the scale grating, the centroid point on QPD1 will oscillate with a period of λ/2 as shown in Eq. (10). The periods for the variations in the displacement outputs of QPD1, i.e., the outputs of θX and θY (the cross-talk errors in θX and θY), can be obtained as:
$$P_{{Z - \theta_{X} }} = P_{{Z - \theta_{Y} }} = \frac{\lambda }{2}$$
(18)

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 PθX and PθY in Fig. 4.

In the presence of ΔZ, the period for the oscillation of the centroid point on QPD2 will be λ/(1 + cosθ) as shown in Eq. (17). The periods for the variations in the displacement output of QPD2, i.e., the output of θZ (the cross-talk error in θZ), can thus be obtained as:
$$P_{{Z - \theta_{Z} }} = \frac{\lambda }{1 + \cos \theta } = \frac{\lambda }{1 + \cos \left( {\arcsin \frac{g}{\lambda }} \right)}$$
(19)

For λ of 405 nm and g of 0.57 μm, PθZ 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 θZ is larger than those in θX and θY. 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.

When an in-plane motion (ΔX or ΔY) is applied to the scale grating, as shown in Eq. (10), there will be no change in the intensities of the interference waves on QPD1, which means that the centroid point will not oscillate and the displacement outputs of QPD1, i.e., the outputs of θX, θY, will not change periodically. This corresponds to the results shown in Figs. 5 and 6. However, on the other hand, the in-plane ΔX or ΔY motion will cause an oscillation of the centroid point on QPD2, resulting a periodic change in the displacement output of QPD2, i.e., the output of θZ (the cross-talk error in θZ) as shown in Eq. (17). The period of the cross-talk error in θZ caused by ΔX or ΔY can be expressed by
$$P_{{XY - \theta_{Z} }} = g$$
(20)

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.

Figure 12 shows a schematic of the improved design. A beam splitter (BS) is added between the scale grating and QWP1 to split the diffracted beams from the scale grating. The reflected beams by the BS are received by the angle assembly for measurement of the angular motions of the scale grating. It can be seen that the angle assembly is out of the optical path of the reference beams and there is no need to employ polarization components for preventing the entry of the reference diffracted beams into the angle assembly. As a result, the cross-talk errors caused by the leakage reference beams in the previous sensor head are expected to be avoided.
Fig. 12

A schematic of the improved design of the sensor head

A detailed optical layout of the improved sensor head is shown in Fig. 13. The angle assembly of the improved design is modified from that of the initial surface encoder by employing not only the zeroth-order and positive first-order diffracted beams but also the negative first-order diffracted beam for improving the performance in θZ measurement [30]. In the displacement assembly, the diffracted laser beams are divided into two groups (groups A and B) by a beam splitter (BS1). One group is directly detected by the PD unit A, while the other is detected by the PD unit B. It should be noted that a 90-degree phase difference is given to each of the beams in the group B with respect to each of them in the group A, respectively. This makes it possible for the surface encoder to recognize the moving direction in the same manner as a conventional optical linear encoder [7]. For compactness of the optical system, two mirrors are located between the PBS and the reference grating in order to keep the optical path difference within the coherence length. Figure 14a shows the mechanical structure of improved six-DOF surface encoder, and Fig. 14b shows a picture of the constructed sensor head. The size of this improved surface encoder is 98 mm (X) × 95 mm (Y) × 28 mm (Z). The size of this improved design is slightly different from the initial one (95 mm (X) × 90 mm (Y) × 25 mm (Z)).
Fig. 13

Optical layout of the improved sensor head

Fig. 14

a 3D view of the mechanical structure of the six-DOF surface encoder. b A photograph of the constructed sensor head

Experiments were carried out to investigate the cross-talk errors in the angular outputs of the improved sensor head. Figure 15 shows the measured cross-talk errors in the outputs of the angle assembly of the improved six-DOF surface encoder, when only ΔX was applied to the scale grating. It can be seen that the cross-talk errors were almost completely reduced from those in the previous design shown in Sect. 2, demonstrating the effectiveness of the improved design of the sensor head.
Fig. 15

Measured cross-talk errors in the outputs of the angle assembly of the improved surface encoder for ΔX

Figure 16 shows the measured cross-talk errors in the outputs of the angle assembly of the improved six-DOF surface encoder, when only ΔZ was applied to the scale grating. Although the cross-talk errors were significantly reduced compared with those in the previous design shown in Sect. 2, periodic error components are still observed in Fig. 16. Figure 17 shows the DFT of the results in Fig. 16, from which the period of the periodic error components is identified.
Fig. 16

Measured cross-talk errors in the outputs of the angle assembly of the improved surface encoder for ΔZ

Fig. 17

DFT of the cross-talk errors in Fig. 16. aθX, bθY, cθZ

After an analysis of the possible reasons, it is identified that the residual periodic cross-talk error components are due to the internal reflection phenomenon of light at the cube surface of BS1. As shown in Fig. 18, the light from LD is divided into two parts. The transmitted part (Beam 1) is projected onto the scale grating and the reflected part (Beam 2) propagates toward the cube surfaceT of BS2. Here, we assume that the beams 1 and 2 have an identical wavefront function of Ein shown as follows:
$$E_{\text{in}} = E_{{{\text{in}}0}}$$
(21)
where E0 is the amplitude of the wave.
Fig. 18

Internal reflection at cube surfaceT in BS1

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 θX and θY motions and QPD D (or QPD E) for measuring the θZ 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).

The wavefront function E2T0 of the part of Beam2T received by QPD C and the wavefront function E2T1 of the part of Beam2T received by QPD D (or QPD E) can be expressed by
$$E_{{2{\text{T}}0}} = m_{0} E_{\text{in}} = m_{0} E_{{{\text{in}}0}}$$
(22)
$$E_{{2{\text{T}}1}} = m_{1} E_{\text{in}} = m_{1} E_{{{\text{in}}0}}$$
(23)
where m0 and m1 are constants determined by the reflectance at the cube surfaceT and the reflection/transmission ratio (R/T) of BS2, as well as the alignment of the components in the sensor head.
Taking into consideration the phase shifts in the scale diffracted beams by ΔZ of the scale grating, the wavefront functions of the scale zeroth-order and first-order diffracted beams can be written as
$$E_{{{\text{s}}0}} = n_{0} E_{{{\text{in}}0}} {\text{e}}^{{i\frac{2\pi }{\lambda }2\Delta Z}}$$
(24)
$$E_{{{\text{s}}1}} = n_{1} E_{{{\text{in}}0}} {\text{e}}^{{i\frac{2\pi }{\lambda }\left( {1 + \cos \theta } \right)\Delta Z}}$$
(25)
where n0 and n1 are constants determined by the diffraction efficiency of the scale grating and R/T of BS2. θ is the diffraction angle shown in Eq. (7).
The wavefront function of a point of the interference fringes generated on QPD C by E2T0 and Es0 can be written as:
$$E_{{{\text{qs}}0}} = E_{{2{\text{T}}0}} + E_{{{\text{s}}0}}$$
(26)
Similarly, the wavefront function of a point of the interference fringes generated on QPD D (or QPD E) by E2T1 and Es1 can be written as:
$$E_{{{\text{qs}}1}} = E_{{2{\text{T}}0}} + E_{{{\text{s}}1}}$$
(27)
The intensity of the interference waves on QPD C Iqs0 and the intensity of the interference waves on QPD D (or QPD E) Iqs1 can thus be obtained as:
$$\begin{aligned} I_{{{\text{qs}}0}} & = E_{{{\text{qs}}0}} \cdot \overline{{E_{{{\text{qs}}0}} }} \\ & = E_{{{\text{in}}0}} \left( {n_{0}^{2} + m_{0}^{2} + 2m_{0} n_{0} \cos \frac{2\pi }{\lambda }2\Delta Z} \right) \\ \end{aligned}$$
(28)
$$\begin{aligned} I_{{{\text{qs}}1}} & = E_{{{\text{qs}}1}} \cdot \overline{{E_{{{\text{qs}}1}} }} \\ & = E_{{{\text{in}}0}} \left\{ {n_{0}^{2} + m_{0}^{2} + 2m_{0} n_{0} \cos \left[ {\frac{2\pi }{\lambda }\left( {1 + \cos \theta } \right)\Delta Z} \right]} \right\} \\ \end{aligned}$$
(29)

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 ΔθX, ΔθY. 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 ΔθZ. The analysis results are well consistent with the results shown in Figs. 16 and 17. It should be noted that the differences between m0, n0 and m1, n1 will make a difference between the amplitude of the periodic cross-talk error component in ΔθX, ΔθY and that in ΔθZ, which are shown in Figs. 16 and 17. ZS, ZR and FS 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, PZθX, PZθY and PZθZ 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

In the optimized design, the BS2 shown in Figs. 12 and 13 are replaced with a plate-type beam splitter (plate-type BS2) for avoiding the influence of the internal reflection shown in the previous section. As shown in Fig. 19, when the light from LD reaches to Surface 1 of the plate-type BS2, the light is divided into two parts. The transmitted part is projected on the scale grating after being refracted at Surface 1 and Surface 2 of the plate-type BS2. The reflected part at Surface 1 propagates toward the outside of the sensor head and will not be received by the angle assembly. Therefore, the interference between Beam2T and the scale diffracted beams, which caused the periodic cross-talk error components in the improved design, can be avoided. Figure 20 shows a close-up picture of the plate-type BS2 and the angle assembly in the optimized sensor head. Surface 1 of the plate-type BS2 transmitted 70% of the incident laser beam, while Surface 2 of the plate-type BS2 was anti-reflective coated at the wavelength of laser beam.
Fig. 19

Employment of a plate-type beam splitter in the optimized design of the sensor head

Fig. 20

A close-up picture of the plate-type BS2 and the angle assembly in the optimized sensor head

Figure 21 shows the outputs of the angle assembly of the optimized six-DOF surface encoder when only a translational motion along Z-directions was applied to the scale grating. Figure 22 shows the DFT of the results in Fig. 21. ZS, ZR and FS were 0.25 nm, 2.49 μm and 0.40 μm-1, respectively. Almost no periodic cross-talk error components are observed in Fig. 21. As shown in Figs. 21 and 22, the cross-talk errors have been reduced to be much less than 0.1 arc-second, which demonstrates the feasibility of the optimized design where the targeted goal of 0.1 arc-second in all the three directions has been achieved. Table 1 shows the results of the previous design, the improved design and the optimized design.
Fig. 21

Measured cross-talk errors in the outputs of the angle assembly of the optimized surface encoder for ΔZ

Fig. 22

DFT of the cross-talk errors in Fig. 16. aθX, bθY, cθZ

Table 1

Cross-talk errors with various optical designs of six-DOF surface encoder for ΔZ

 

θX arc-second

θY arc-second

θZ 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 θX, θY and θZ, 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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Copyright information

© International Society for Nanomanufacturing and Tianjin University and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Finemechanics, Graduate School of EngineeringTohoku UniversitySendaiJapan
  2. 2.Graduate School at ShenzhenTsinghua UniversityShenzhenChina

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