Two-channel six degrees of freedom grating-encoder for precision-positioning of sub-components in synthetic-aperture optics

Kangning Yu; Junhao Zhu; Weihan Yuan; Qian Zhou; Gaopeng Xue; Guanhao Wu; Guanhao Wu; Xiaohao Wang; Xinghui Li; Xinghui Li; Xinghui Li

doi:10.1364/OE.427307

1. Introduction

Large-scale synthetic aperture optics (SAO) is very important for astronomical observations [1–3] and high-energy laser-physics research [4–6]. The European Extremely Large Telescope (E-ELT), the largest optical telescope under construction in the world, has a primary mirror that consists of 798 small hexagonal lenses [7]. This telescope will be able to accelerate space-exploration significantly. The large parabolic mirror, which consists of several small PCGs (as proposed by researchers from NIF and HiPER of Europe [4,6]), is the key component to produce the compression of the laser and is particularly important for basic research in physics. The performance of the SAO, however, is limited by the positioning accuracy of its sub-components. The Southern African Large Telescope (SALT), the largest optical telescope in the southern hemisphere [8]. Its observation performance suffers significantly, however, when there are deviations between the sub-mirrors due to thermal and gravity changes [9]. When used in high-energy laser amplifiers, the deviation in the position of the sub-components of the large-scale synthetic aperture PCGs can reduce the laser energy [5]. Therefore, it is highly desirable to position the sub-components of an SAO with multiple DOFs very accurately.

At present, the technologies that can meet the position-accuracy requirements include capacitive sensors [10,11], laser interferometers [12,13], geometric optical measurement [14–17], optical heterodyne interferometry [18–20], and encoder technology [21–25]. A multi-DOF measurement-system, which consists of 168 capacitive sensors [26], is used for the real-time detection of the sub-mirrors position of the Keck Telescope. Even though the whole measurement system is very accurate and stable [27], it is complicated and sensitive to both temperature and humidity [28]. The laser interferometer can be used to detect the position of sub-components in an SAO with high resolution and high precision. Liu’s group proposed a six-DOF measurement-system [29] that used a dual laser interferometer. Its linear displacement accuracy could reach 10 nm, while the angular displacement accuracy reached 3 milli-arcseconds. However, the measurement accuracy of the laser interferometer is easily affected by the environment, and the equipment is complex and large. This limits applications, especially when good real-time performance is required. Feng’s group proposed a geometrical optical system [15] which can simultaneously measure the six-DOF motion errors of the linear guide. The standard deviation of the linear motion of ±100µm is 0.5µm, and the standard deviation of the angular motion of ±100” is less than 1.0”. However, the large volume of the measuring system makes it difficult to apply to synthetic optical elements. Badami’s group proposed a new optical heterodyne encoder [19], which can achieve measurement accuracy up to subnanometer at a speed of 8m/s. However, the six-DOF motion measurement requires a combination of multiple encoders, which will cause system complexity and Abbé error problems. Furthermore, the grating interferometer could measure the position of objects with multiple DOFs. It is compact and very accurate, which makes it suitable for the required high-precision real-time positioning of the sub-components in an SAO.

The multi-DOF grating interferometer system is becoming an important component for high precision positioning [30,25]. From 2005, Gao’s group [31–33] performed three-DOF displacement measurements and three-DOF angle measurements with a planar grating, which featured a displacement resolution of 10 nm and an angle resolution of 0.01”. From 2015, Hsieh’s group proposed a six-DOF measurement system, which was based on a planar grating with a displacement- and angle-measurement - resolution of 2 nm and 0.05 µrad [34], respectively. But the optical structure was complex and the measurement range was small in the vertical direction (due to focal length limits of the lens). From 2013, Li’s group developed a six-DOF grating encoder [35], which consisted of a sensor head and a nanometer planar grating. The grating period is 0.57 µm which was a high precision and uniformity micron-order planar grating fabricated by a interference lithography process [36–39]. The resolution of the linear-displacement measurement of the grating encoder was up to 2 nm in the X-, Y-, and Z-directions. The resolution of the angle measurement in the θ_x-, θ_y-, and θ_z-directions reached 0.1”, 0.1”, and 0.3”, respectively. However, the relative position deviation of the two sub-components could not be detected because only one laser beam was used in the grating encoder.

In this paper, a new two-channel six-DOF grating encoder is proposed, which can detect the position of two sub-components of an SAO simultaneously. The two-channel six-DOF measurement can be obtained by simply combining a three-DOF displacement sensor and a three-DOF autocollimator who share the same laser source (which contains two beams). In addition, a new optical layout was used to reduce stray laser-transmission in the grating encoder, which can reduce the impact of any imperfections of the optical elements. In this study, a prototype of a two-channel six-DOF grating encoder was designed and fabricated, and a number of performance tests were carried out using high-precision actuators.

2. Design of the two-channel six-DOF grating encoder

Figure 1 illustrates the operating principle of a SAO which is assisted by a two-channel six-DOF grating encoder. The SAO consists of the parabolic mirrors Cell1 and Cell2. Note that only two cells are shown for clarity. The SAO focuses the incident beam optically. However, the performance of the SAO decreases if the foci of Cell1 and Cell2 are not aligned. The position of the foci of the two cells depends on the position of the cells, which are adjusted so that their focal points coincide before the SAO is used. However, the performance of the SAO is reduced by the six-DOF motions of the cells, which can be affected by changes in gravity, temperature, and wind. The six-DOF motions include the translational displacement motions Δx, Δy, and Δz along X-, Y-, and Z-directions as well as the angular motions Δθ_x, Δθ_y, and Δθ_z about X-, Y-, and Z-directions, respectively. The movements of two cells are detected by the two-channel six-DOF grating encoder and the movement information is sent to the control unit. Therefore, the SAO cells are set to the calibrated position if the cells deviate from the original position, while the SAO is used.

Fig. 1. The two-channel six-DOF grating encoder used with the SAO.

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A prototype of a two-channel six-DOF grating encoder was designed to measure motions of two SAO cells involving six axes simultaneously. The grating encoder consists of a sensor head and two scale gratings. The scale gratings are mounted on SAO cells respectively, while the sensor head is mounted on a fixed bracket. The measurement resolution and range of the grating encoder are the highest priority during the first stage of research. The target resolution for the grating encoder was set to 1 nm for translational displacements and 0.02 arc-second for angular motions. The measurement range of the grating encoder should be larger than the moving range of the SAO cells.

The first step of the design work was to establish an operating principle that can easily realize the desired single six-DOF measurement. Because the operating principle of the dual six-DOF measurement is based on that of single-six-DOF measurement. As can be seen from Fig. 2, the optical sensor head of the single-channel six-DOF grating encoder consists of both a displacement-assembly and an angle-assembly. The displacement assembly, which relies on the operating principle of a three-axis displacement sensor, was used to measure the translational motions Δx, Δy, and Δz. The angle assembly, which relies on the operating principle of a three-axis autocollimator, was used to measure the angular motions Δθx, Δθy, and Δθz. The displacement- and angle-assemblies were then combined to form a simple sensor-configuration that shares the same laser source.

Fig. 2. Operating principle of the single-channel six-DOF grating encoder.

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As shown in Fig. 2, a collimated beam is emitted from the laser source, which consists of a laser diode with the wavelength λ and a collimating lens. The collimated beam passes through the beam-splitter prism 1 (BS1) and is split into two beams, which are projected onto the scale grating and the reference grating, respectively. The scale grating and reference grating have the same structure, with the same grating period g in the X- and Y-directions. The X- and the Y-directional first (1st) order diffraction beams are generated at each grating surface with a diffraction angle of φ. The diffraction beams from the reference grating are bent at BS1, while the beams from the scale-grating are transmitted through BS1. Then two groups of diffraction-beams interfere with each other and the superimposed signals (X+1, X−1, Y+1, Y−1), which are detected by the photoelectric detector (PD) units A (AX+1, AX−1, AY+1, AY−1). The current signals I_AX₊₁ and I_AX₋₁ from PD AX+1 and PD AX−1 are cosine functions, which contain the translational displacements Δx and Δz. Similarly, the current signals I_AY₊₁ and I_AY₋₁ from PD AY+1 and PD AY-1 are cosine functions, which contain the translational displacements Δy and Δz. The translational displacement motions Δx, Δy, and Δz can be derived from the above current signals. The period of Δx and Δy along X- and Y-directions is g, and the period of Δz is λ/(1+cosφ) along Z-direction. However, the direction of motions cannot be identified from the cosine function because of its nature. This problem, however, can be solved using an improvement of the optical structure (see below).

In addition, as shown in Fig. 2, the diffraction beams coming from the scale grating to the angle assembly are bent at beam splitter 2 (BS2). Each diffraction beam is received by the auto-collimating unit, which consists of a quadrant photoelectric detector (QPD) and a collimator objective (CO) with a focal length f. The QPD is placed in the focal plane of the COs to detect the motion of the light spot focused on it. The zeroth (0th) order and first negative (−1) order diffraction beams, which were generated at the scale grating, are detected by QPD A and QPD B, respectively. The current outputs of the four cells of the QPD A and QPD B are denoted I_Ai (i=1, 2, 3, 4) and I_Bi (i=1, 2, 3, 4), respectively. The position of the light spot on QPD A and QPD B is characterized by Δh_A, Δv_A, Δh_B, and Δv_B, which can be calculated as follows [40]:

(1)$$\Delta {h_A} = {k_1}\frac{{{I_{A1}} + {I_{A4}} - {I_{A2}} - {I_{A3}}}}{{{I_{A1}} + {I_{A2}} + {I_{A3}} + {I_{A4}}}}$$

(2)$$\Delta {v_A} = {k_2}\frac{{{I_{A1}} + {I_{A2}} - {I_{A3}} - {I_{A4}}}}{{{I_{A1}} + {I_{A2}} + {I_{A3}} + {I_{A4}}}}$$

(3)$$\Delta {h_B} = {k_3}\frac{{{I_{B1}} + {I_{B4}} - {I_{B2}} - {I_{B3}}}}{{{I_{B1}} + {I_{B2}} + {I_{B3}} + {I_{B4}}}}$$

(4)$$\Delta {v_B} = {k_4}\frac{{{I_{B1}} + {I_{B2}} - {I_{B3}} - {I_{B4}}}}{{{I_{B1}} + {I_{B2}} + {I_{B3}} + {I_{B4}}}}$$

The coefficient k₁, k₂, k₃, and k₄ depend on the properties of the QPD, which can be obtained experimentally.The initial position of the light spot is assumed to be in the center of the QPDs. The angular motions Δθ_x, Δθ_y, and Δθ_z of the scale grating can be calculated as follows [41]:

(5)$${\theta _x} = \frac{{\Delta {v_A}}}{{2f}}$$

(6)$${\theta _y} = \frac{{\Delta {h_A}}}{{2f}}$$

(7)$${\theta_z} = \frac{g}{{f\lambda }}\left( {\Delta {v_A} - \Delta {h_B}} \right)$$

The second step is to design the optical layout of the two-channel six-DOF grating encoder which is shown in Fig. 3(a). It is designed according to the operating principle shown in Fig. 2. The wavelength (λ) of a laser diode and the period (g) of the grating were set as 0.66 µm and 1 µm, respectively. The 1st order diffraction angle of φ was (calculated) 41.3°.

Fig. 3. Optical layout of the two-channel six-DOF grating encoder. (a) Schematic of the grating encoder. XZ view. (b) Schematic of the grating encoder. 3D view. (c) Diffraction beams generated at the scale-grating module. (d) 0^th order and X-direction of the ±1^st order diffraction beams are guided to the QPDs in the angle assembly. (e) Diffraction beams are guided to the PD units A and PD units B of the displacement assembly.

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Figure 3(b) shows a 3D schematic of the two-channel six-DOF grating encoder, which correlates with the 2D view in Fig. 3(a). The grating encoder consists of five modules, which include a laser-source module, reference grating module, scale grating module, displacement measurement module, and angle-measurement module. For better clarity, only the main light-paths are drawn in Fig. 3. The optical elements in Fig. 3(b) and Fig. 3(a) are consistent with respect to each other.

Many improvements over Fig. 2 are shown in Fig. 3. As shown in Fig. 3(a), the laser diode 2 (LD2) was added so that the dual six-DOF measurement can be obtained by sharing the same optical structure for the lasers of LD1 and LD2. The polarizer 1 was added so that the same polarization direction of the beam can be obtained. The quarter-wave plate 1 (QWP1) and quarter-wave plate 2 (QWP2) were added to adjust the polarization direction of the beam. The collimated beams from LD1 and LD2 pass through the polarizer 1, with a polarization direction of 45° with respect to the horizontal direction. Subsequently, they are divided into two beams using a polarizing beam splitter prism (PBS). The two beams are projected onto the scale grating and the reference grating, respectively. The 0th order, as well as the X- and Y-directional 1st order diffraction beams are generated at each grating surface, with a diffraction angle of φ. They are bent as parallel beams by the prism units - see Fig. 3(c). The diffraction beams from the reference grating redirected using the PBS and the beams from the scale grating pass through the PBS. Then, two groups of diffraction beams are projected into the displacement assembly, which has many improvements.

Figure 3(d) shows an improved schematic of the displacement assembly with the PD units A and B, with the goal to obtain the direction for translational-displacement motions Δx, Δy, and Δz. Each of the PD units consists of four individual two-cell photoelectric detectors, where two separate cells are used to replace the quadrant photoelectric detector in Fig. 2. The interference signals of the diffraction beams from the reference grating and the scale grating are detected by the PD units. There is a 90° phase difference for the outputs between the PD units A and PD units B because of the added QWP3. Therefore, the directions of Δx, Δy, and Δz can be identified by the grating encoder. The current outputs of the PD units A and B are I(i=1) AX+1, I(i=1) AX−1, I(i=1) AY+1, I(i=1) AY−1, I(i=2) AX+1, I(i=2) AX−1, I(i=2) AY+1, and I(i=2) AY−1, where the numbers 1 and 2 denote LD1 and LD2, respectively. The translational displacement motions Δx_i, Δy_i, and Δz_i (i=1,2) can be calculated as follows [42]:

(8)$${\Delta }{{x}_{i}} = {{{{k}_{l}}{g}} \over 2} + {{g} \over {4{\pi }}}\left\{ {{\textrm{arctan}}\left( {{{{I}_{{BX} + 1}^{i}} \over {{I}_{{AX} + 1}^{i}}}} \right) - {\textrm{arctan}}\left( {{{{I}_{{BX} - 1}^{i}} \over {{I}_{{AX} - 1}^{i}}}} \right)} \right\}$$

(9)$${\Delta }{{y}_{i}} = {{{{k}_{m}}{g}} \over 2} + {{g} \over {4{\pi }}}\left\{ {{\textrm{arctan}}\left( {{{{I}_{{BY} + 1}^{i}} \over {{I}_{{AY} + 1}^{i}}}} \right) + {\textrm{arctan}}\left( {{{{I}_{{BY} - 1}^{i}} \over {{I}_{{AY} - 1}^{i}}}} \right)} \right\}$$

(10)$$\begin{array}{l} {\Delta }{{z}_{i}} = \frac{{{{k}_{n}}{\lambda }}}{{2({1 + {\cos}{\theta }} )}} + \frac{{\lambda }}{{8{\pi }({1 + {\cos}{\theta }} )}}\left\{ {{\textrm{arctan}}\left( {\frac{{{I}_{{BX} + 1}^{i}}}{{{I}_{{AX} + 1}^{i}}}} \right) + } \right.\\ \left. {{\textrm{arctan}}\left( {\frac{{{I}_{{BX} - 1}^{i}}}{{{I}_{{AX} - 1}^{i}}}} \right) + {\textrm{arctan}}\left( {\frac{{{I}_{{BY} + 1}^{i}}}{{{I}_{{AY} + 1}^{i}}}} \right) + {\textrm{arctan}}\left( {\frac{{{I}_{{BY} - 1}^{i}}}{{{I}_{{AY} - 1}^{i}}}} \right)} \right\} \end{array}$$

The coefficient k_l, k_m, and k_n represent the period number for motions in the X-, Y-, and Z-directions.

Figure 3(e) shows a schematic of the improved angle assembly. A beam splitter plate (BSP) is used in the angle assembly instead of BS2 in Fig. 2, which can reduce the interference impact due to the reflected beams from the inner surface of the cube BS2. In addition, the BSP is placed between the scale grating and the PBS, which can reduce the interference impact caused by imperfections of the polarizer as observed in previous research. Both +1st and −1st order diffraction beams for the X-direction are involved in the calculation of the reduction of environmental impact. COs with a short focal length are used for the compact size of the angle assembly, and the focal lengths are 12.5 mm and 4.4 mm for ±1st and 0th order diffraction beams, respectively.

Figure 4 shows a picture of a prototype sensor head for the two-channel six-DOF grating encoder. Several details should be paid attention to when installing the optical elements. Firstly, the polarization direction of the laser beams and the transmission axes of polarizer should be adjusted to an angle of 45° (with respect to the horizontal direction), so that the same power of two beams, which was generated at the PBS, can be obtained. Then, the prism units, which are situated next to the reference grating and the scale grating, should be adjust carefully to minimize the number of interference fringes. Moreover, the position of the QPD should be adjusted to coincide with the focal plane of the COs because of its sensitivity with respect to any motions of the light spot. Meanwhile, to obtain the best linearity of the QPD output, the beam should be focused at the center of the QPD.

Fig. 4. Photograph showing the prototype sensor head of the two-channel six-DOF grating encoder.

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3. Testing of the two-channel six-DOF grating encoder

Figure 5(a) shows the experimental setup to test the grating encoder using a measurement of the dual six-DOF translational and angular motions. The setup consists of the sensor head of the two-channel six-DOF grating encoder, a five-DOF actuator, a dual-frequency laser- interferometer, an autocollimator, and a circuit module. The target grating was mounted on the five-DOF actuator to serve as a scale grating (which is driven by the five-DOF actuator). The dual-frequency laser-interferometer can measure the three-axis translational displacement motions of the target grating with high precision. The result can be used as reference for the calculated output of the grating encoder. The commercial dual-frequency laser-interferometer with an accuracy of 160 nm, a resolution of 0.1 nm, and a measurement range of 10 m. Similarly, the commercial autocollimator can determine the two-axis angular motions of the target grating with high precision. The results of the autocollimator can be used as references for the calculated outputs of the grating encoder. The circuit module is used to convert the current signal into a voltage signal and send it to the data acquisition (DAQ) card.

Fig. 5. (a) Experimental setup. (b) Scale grating, mounted on a 5-DOF actuator.

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Figure 5(b) shows the layout of the five-DOF actuator in more detail. A square XY planar grating, with a period of 1 µm in the X- and Y-directions and a length of 15 mm on each side, was used as the target grating for the experiment. The target grating was mounted on the five-DOF actuator, which consists of three linear actuators and a two-axis PZT tilt actuator to generate the three-axis translational displacement motions (Δx, Δy, Δz) and the two-axis angular motions (Δθx and Δθz). Each open-loop linear actuator has a one-way repetition accuracy of 0.25 µm, and a measurement range of 25 mm. The closed-loop controlled two-axis tilt PZT actuator has a resolution of 0.05 µrad, a linearity of 0.5%, and a measurement range of 8mrad. To generate the angular motion Δθx, the two-axis PZT tilt actuator was rotated 90° about the X-axis. The position of the sensor head was adjusted using the Y-axis manual stage to ensure alignment with the target grating. In addition, the five-DOF actuator was mounted on the Z-axis manual stage, which was used to realize a suitable position of the target grating.

First, the resolution of the two-channel six-DOF grating encoder was measured. Figure 6 shows the results for Δx, Δy, Δz, Δθx, Δθy, and Δθz. The results on the left and right are from probe 1 and probe 2 in the sensor head. The target grating was moving 100 µm in the X-, Y-, and Z-directions at a velocity (v) of 1 µm/s. Furthermore, the sampling frequency (f_s) of the DAQ card in each motion direction were 20000 Hz, 20000 Hz, and 10000 Hz, respectively. The theoretical displacement resolution (R_t) can be calculated as follows:

(11)$${{R}_t} = {v \over {{f_s}}}$$

Fig. 6. (a) and (b) indicate the displacement resolutions for Δx, Δy, Δz of probes 1 and 2. (c) and (d) indicate the angular resolutions of Δθx, Δθy, Δθz of probe 1 and probe 2.

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The theoretical resolutions were (calculated) 0.05 nm, 0.05 nm, and 0.1 nm in the X-, Y-, and Z-directions, respectively, which are marked as green lines in Fig. 6. The actual displacement resolution (R_a) can be calculated as follows:

(12)$${R_a} = \overrightarrow {{a_{i + 1}}} - \overrightarrow {{a_i}}$$

Here, a=(Δx, Δy, Δz) denotes the translational displacements in three directions, as shown by the blue lines in Fig. 6. The character i indicates the i-th sample point. The actual displacement resolution is shown as red line in Fig. 6(a). It can be seen that the actual displacement resolution errors of the grating encoder are mostly within ±0.02 nm of the theoretical resolution shown in the orange lines in Fig. 6(b).

Figure 6(c) and (d) shows the angular resolutions of Δθx, Δθy, and Δθz from probe 1 and probe 2 in the sensor head. The resolution of the grating encoder to detect the angular motions Δθx, Δθy, and Δθz were determined by applying cosine tilt motions to the target grating using the two-axis PZT tilt actuator. The cosine motions (amplitude 0.02”) can be distinguished (using both probes) from the sensor head in the θx-, θy-, and θz-directions.

Figure 7 shows the linear displacement residuals for the displacement outputs of probes 1 and 2 of the sensor head. To determine the linearity of the grating encoder, the target grating was moved 4 mm, 10 mm, and 1 mm in the X-, Y- and Z-directions at a velocity of 5 µm/s. In addition, the motions of the target grating were detected using the sensor head and the dual-frequency laser-interferometer simultaneously. The results of the grating encoder were compared with those of the dual-frequency laser-interferometer. The blue line in the displacement results combines the output of the grating encoder and the interferometer. The horizontal axis in the figure represents the output of the dual-frequency laser-interferometer, while the left vertical axis represents the output of the grating encoder. In addition, the figure shows that the residual error is ±1 µm within a measurement range of 4 mm in the X-direction, ±4 µm within a measurement range of 10 mm in the Y-direction, and ±0.2 µm within a measurement range of 1 mm in the Z-direction. The linearity of the grating encoder in the X-, Y-, Z-directions was calculated 0.025%, 0.04%, and 0.02%, respectively. The residual lines show the periodic errors in all three directions, which is caused by the nonlinearity of the ball screw in the linear actuators. This is because the period of the residuals is the same as the period of the ball screw. It can be found that the periods for the ball screw in the linear actuator in the X-, Y-, and Z-directions are 1 mm, 1 mm, and 0.4 mm, respectively, which is exactly equal to the periods of the residuals. It is important to note that the residual errors for Δx, Δy, and Δz can be reduced by using linear actuators with higher precision.

Fig. 7. (a) Linear displacement residuals for the displacement outputs of probes 1. (b) Linear displacement residuals for the displacement outputs of probe 2.

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Figure 8 shows the angular residuals for Δθx, Δθy, and Δθz, which were detected by probe 1 and probe 2 in the sensor head. In the test, the target grating was moved by the two-axis tilt PZT actuator to rotate 25” about the X-, Y-, and Z-axes at a velocity of 1"/s. The angular motions of the target grating were detected by using the sensor head and autocollimator simultaneously.

Fig. 8. (a) Angular residuals of probe 1. (b) Angular residuals of probe 2.

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The blue line of angular results combines the output of the grating encoder and the autocollimator. The horizontal axis of the figure represents the output of the autocollimator, while the left vertical axis represents the output of the grating encoder. The angular residuals of Δθx, Δθy, and Δθz are 0.35", 0.35”, and 0.21”, respectively, within a stroke of 25”. These numbers are better than previously reported research (2.2”, 1.4”, 2.5” about Δθx, Δθy, Δθz) [35].

The last measurement was performed to investigate the cross-talk errors in the grating encoder outputs, when the target scale grating was driven by three linear actuators in the X-, Y-, and Z-axes, respectively. The cross-talk errors about the θx-, θy-, and θz-axes are not mentioned because of their negligible effect on the other five directions. The outputs of the dual six-DOF of the grating encoder are required during movement. To reduce the cross-talk errors caused by the imperfections of the linear actuators themselves, only a short movement-range of 5 µm was used in each direction. Figures 9(a) and (b) show the cross-talk errors, when the target grating was moved in the X-direction. The variation outputs for Δy, Δz, Δθx, Δθy, and Δθz indicate the cross-talk errors of the grating encoder during the movement the X-direction. The output Δx of the grating encoder in the X-axis is not shown for clarity. The cross-talk errors are approximately 14 nm, 30 nm, 0.05”, 0.1”, and 0.06” in the Δy, Δz, Δθx, Δθy, and Δθz directions, respectively. The cross-talk errors of the experimental results is in the same order of magnitude as that of the previous work [43].

Fig. 9. (a) and (b) are the cross-talk errors of probe 1 and probe 2 when applied to Δx. (c) and (d) are the cross-talk errors of probe 1 and probe 2 when applied to Δy. (e) and (f) are the cross-talk errors of probe 1 and probe 2 when applied to Δz.

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The cross-talk errors for Δy and Δz are linear errors due to misalignment between the axis of the sensor head of the grating encoder and the axis of the target grating. It is important to note that this linear error can be reduced by using a more careful alignment of the optics and compensation [42]. The cross-talk errors for Δθx, Δθy, and Δθz are nonlinear errors, which are likely to be caused by the interference with the beams from the BSP. A beam from the target grating, which is reflected at two surfaces of the BSP, can form two beams which pass through the CO and interfere with each other at the QPD. The following calculation verifies the validity of the detection. The angular cross-talk error shows a cosine function about 7.5 periods within a stroke of 5 µm. The period of the cosine function is (calculated) 667 nm, which is approximately equal to the period g (g=660 nm) of the interference signal. This nonlinear error can be reduced by removing any stray beams. Similarly, as shown in Figs. 9(c) and (d), the cross-talk errors are approximately 4 nm, 20 nm, 0.05”, 0.04”, and 0.03” in the Δx, Δz, Δθx, Δθy, and Δθz directions, respectively, during the movement in the Y-direction. In addition, the cross-talk errors can also be reduced using the above method.

As shown in Figs. 9(e) and (f), the cross-talk errors for Δx and Δy have an additional periodic component (with a 10 nm amplitude) based on the linear variation. These cross-talk errors are due to interpolation errors from the translational displacement Δz. This is because of the equality of the period of cross-talk errors for Δx and Δy with Δz. It can be reduced by a more careful alignment of both the optics and compensation. Furthermore, in Figs. 9(e) and (f), the cosine amplitudes of the cross-talk errors for Δθx, Δθy, and Δθz are larger than those in Figs. 9(a), (b), (c), and (d). Particularly, the difference in the cross-talk errors between probe 1 and probe 2 is unexpected. Firstly, the cross-talk errors are expected to be caused by the interference of the beams from the BSP. Then, the amplitude of the cosine function is very large because the motion for the Z-direction can cause stronger interference. Finally, the different cross-talk amplitudes between probe 1 and probe 2 are caused by the different intensities between the stray beams in two probes. Fortunately, the cross-talk errors for Δθx, Δθy, and Δθz can be reduced by removing the stray beams.

4. Conclusion

We designed and investigated a two-channel six-DOF grating encoder, which consists of two planar gratings and an optical sensor head with a 660 nm wavelength laser. The six-DOF measurement can be performed through a simple combination of a three-axes displacement sensor with a three axes autocollimator. This can be done by sharing the same laser source which includes two beams. The grating period and the grating size of the planar grating were 1 µm and 15 mm (X)×15 mm (Y), respectively. They determine both the resolution and the measurement range of the grating encoder.

Experiments were carried out to measure the performance of the two-channel six-DOF grating encoder. It was confirmed that the grating encoder could identify the cosine motion with an amplitude of 0.02” in the θx-, θy-, and θz-directions, and the numerical displacement resolutions were 50 pm, 50 pm, and 100 pm in the X-, Y-, and Z-directions, respectively. The peak-to-valley amplitudes of the residuals for the displacement were ±1 µm, ±4 µm, and ±0.2 µm along the stroke of 4 mm, 10 mm, and 1 mm in the X-, Y-, and Z-directions, respectively. The linearity of the displacement outputs was 0.025%, 0.04%, and 0.02%. The peak-to-valley amplitudes of the residuals of the angular outputs were ±0.15”, ±0.15”, and ±0.12” along the stroke of 25” in the θx-, θy-, and θz-directions, respectively. The linearity of the angular outputs was 0.6%, 0.6%, and 0.48%. The error elements can be decreased by reducing the stray laser and replacing the high-precision actuators. The cross-talk errors of the grating encoder were also investigated. The periodic cross-talk error components of the angular outputs, which were due to the stray laser interference, were determined. The dominant errors include: grating period deviation, laser diode instability, optical element imperfection, and crosstalk error. Firstly, we calibrate each region of the grating to reduce the error caused by the grating period deviation. Secondly, we use a constant current source to control the laser diode, and control the environment constant temperature to ensure the stability of the laser diode. Thirdly, we use algorithms to reduce the spectral imperfections of optical elements. Finally, we reduce the crosstalk error by compensation. The uncertainty of the proposed system is 0.145µm in three displacement directions and 0.17” in three angle directions. Six identical experiments were carried out in the Y-direction when the stroke was 10mm, and the calculated uncertainty of the system repeatability accuracy was 0.145µm. Similarly, the uncertainty analysis of the other five directions was carried out. However, the uncertainty analysis only involves the repeatability of measurement, and the uncertainty analysis caused by environmental interference and circuit noise will be carried out in the future research work. In addition, the experimental results show that the measurement performance of the two probes is the same, and the measurement difference between the two probes is in the submicron and subarc second levels.

Future work could include the reduction of the cross-talk-error components in the angular outputs, using actuators with higher precision, measuring the performance of the grating encoder for a larger scale, and minification of the grating encoder. Furthermore, the stability of the grating encoder over a long period of time is very important, and its stability should be tested when it is integrated with the SAO.

Funding

Shenzhen Fundamental Research Program (JCYJ20170817160808432); National Natural Science Foundation of China (52005291, 61905129); Natural Science Foundation of Guangdong Province (2018A030313748); Tsinghua University (QD2020001N).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Two-channel six degrees of freedom grating-encoder for precision-positioning of sub-components in synthetic-aperture optics

Abstract

1. Introduction

2. Design of the two-channel six-DOF grating encoder

3. Testing of the two-channel six-DOF grating encoder

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (12)

Optics Express