## Abstract

A three-degrees-of-freedom measurement system based on the Faraday effect is proposed for simultaneously measuring two-dimensional straightness errors and their position. Thanks to the Faraday effect of the Faraday rotator, the direction of a linearly polarized beam can be changed by 90° when the linearly polarized beam passes through the same Faraday rotator back and forth twice. A novel optical configuration is designed that can integrate the interferometry and position-sensitive detection technology ingeniously and put their advantages together. The measurement principle is described in detail. The influence of angle error of the semitransparent mirror on straightness measurement is discussed. To verify the feasibility of the proposed system, the experimental setup for measuring three degrees of freedom was constructed, and a series of experiments were carried out.

© 2020 Optical Society of America

## 1. INTRODUCTION

Among the six degrees of freedom of an object in three-dimensional space, the parameter of straightness plays an important role in evaluating the performance of precision stages or guide ways, and straightness error has a great influence on their motion accuracy [1]. The traditional straightness measurement methods cannot satisfy the requirement of high-efficiency and high-accuracy measurement, such as electronical gradienter, straight-edge method or taut wire [2,3]. Since the laser was invented, it has been widely used to realize precision straightness measurement for its advantages of high coherence and good directivity [4–6].

Although traditional laser straightness interferometers can realize high-precision straightness measurement, such as Renishaw and Agilent laser straightness interferometers, they only can measure a one-dimensional straightness error that is a horizontal or vertical straightness error, and they seldom give the position of the measured straightness error [7,8]. To solve this problem, Chen *et al* proposed a laser interferometer for measuring straightness and its position based on heterodyne interferometry [9]. However, due to using a Wollaston prism to split a laser beam in these straightness interferometers, the measurement length of the measured straightness is limited by the divergent angle of the Wollaston prism. Based on lasers’ advantage of good directivity, position-sensitive detection is widely used to realize straightness measurement by using a position-sensitive detector (PSD) or four-quadrant detector (QPD), which have the merit of simultaneous measurement of horizontal and vertical straightness [10,11]. However, limited to the photosensitive area of PSD or QPD, the position-sensitive detection cannot be used to realize large-range displacement measurement, such as displacement calibration of long-travel linear guide ways [12].

In order to meet the important requirement of precision instruments applicable to multiple degrees of freedom measurement, many multiple degrees of freedom measurement methods have been proposed based on laser technology [13–15]. For example, Feng *et al* introduced a simultaneous measurement system of six-degrees-of-freedom geometric motion errors, which utilizes a QPD and a photodetector (PD) for measuring straightness error and its position, respectively [16]. Chen *et al* designed a simultaneous measurement system of six-degrees-of-freedom error parameters that uses two PDs to measure the optical path difference between two optical path lengths to get the vertical straightness error and its position and obtains horizontal straightness error by using a QPD [17]. Yu *et al* developed a simultaneous measurement system of multi-degrees-of-freedom, which obtains the straightness error by using PSD, and the position error is measured by using QPD [18]. In view of the above research, the realization of multi-degrees-of-freedom measurement usually adopts position-sensitive detection and interferometry; the position-sensitive detection can realize two-dimensional straightness errors measurement simultaneously, and the interferometry can realize large-range and precision displacement measurement.

In our previous work, we proposed an orthogonal return method for a linearly polarized beam based on the Faraday effect, which has been verified to apply in laser heterodyne interferometers for precision displacement measurement [19,20]. Based on this orthogonal return method, in this paper, a three degrees-of-freedom system for measuring two-dimensional straightness errors and their position is proposed based on position-sensitive detection and interferometry, which put their advantages together. The measurement optical configuration is designed. The two-dimensional straightness measurement is realized by using a PSD, and the position of the straightness error is measured by using laser heterodyne interferometry.

## 2. CONFIGURATION

Figure 1 shows the proposed optical configuration for measuring two-dimensional straightness errors and their positions. Based on laser position-sensitive detection, a PSD is used for measuring the horizontal and vertical straightness. The position of the straightness error is detected by using heterodyne interferometry, and the interferometer for the measuring position is realized by using the orthogonal return method for a linearly polarized beam based on the Faraday effect [19]. One merit of the interference optical configuration is that correct return of the measurement beam is ensured when large lateral displacement of a measured object occurs, and the lateral displacement is either the straightness error or the measured object. As is shown in Fig. 1, a stabilized dual-frequency He–Ne laser emits an orthogonally linearly polarized laser beam. The beam is divided by a polarizing beam splitter (${{\rm PBS}_1}$) into a reference beam and a measurement beam: the reference beam enters the reference arm. The reference beam reflected by ${{\rm PBS}_1}$ transmits a quarter-wave plate (${{\rm QP}_1}$) and projects onto a plane reflective mirror (RM); then the reference beam is reflected by the RM and transmits the ${{\rm QP}_1}$ and ${{\rm PBS}_1}$ sequentially. The measurement beam enters the measurement arm and passes through a Faraday rotator (FR) with a rotational angle of 45°, then passes through ${{\rm PBS}_2}$, which is fixed with the same rotational angle. The measurement beam transmits ${{\rm QP}_2}$ and incidents onto a hollow corner cube (HCC) prism, the measurement beam is reflected by the HCC, then transmits ${{\rm QP}_2}$ again, reflected by ${{\rm PBS}_2}$ and incidents onto a semireflective mirror (SR). Here, the measurement beam is split into a transmitted part and a reflected part; the transmitted part projects onto a PSD, which is used for straightness measurement, and the reflected part is reflected back along the incoming optical path and transmits FR again. Then, the polarization direction of measurement beam changes 90° due to the Faraday effect of the FR, so the returned measurement beam is reflected by ${{\rm PBS}_1}$. Finally, the returned beam from the reference arm and measurement arm transmit the same polarizer (P) and project onto a PD; then the measurement signal is generated in the PD. In this configuration, the reference signal is provided from the rear of the laser head. Through processing the reference signal and the measurement signal from the PD, the position of the HCC can be detected, and by acquisition and processing the position of the laser spot on the PSD, the two-dimensional straightness errors can be obtained.

A traditional straightness measurement interferometer by using a Wollaston prism can realize straightness error measurement, but the position of the straightness error cannot be measured simultaneously, and only one-direction straightness can be measured each time. And compared with a traditional straightness interferometer, the proposed system not only can realize a two-dimensional straightness error measurement, but also can realize the position measurement of the two-dimensional straightness errors simultaneously.

## 3. MEASUREMENT PRINCIPLE

#### A. Measurement Principle of Straightness

Figure 2 shows the straightness error measurement schematic of the proposed method, including a horizontal straightness error and a vertical straightness error. As the measurement principle of horizontal straightness error is the same as that of the vertical straightness error, the horizontal straightness error measurement is taken as an instance to introduce the measurement principle. As is shown in Fig. 2, when the HCC moves along with the measured object, it has a horizontal straightness deviation from the position H0 to H1, the laser spot on the PSD deviates from the initial red spot position to the latter blue spot position, and the position change of laser spot on the PSD indicates the straightness error of the measured object.

In the optical configuration shown in Fig. 1, because the ${{\rm PBS}_2}$ and the PSD are placed with the rotational angle of 45° in clockwise direction, the position-sensitive detection coordinate system ($x - o - y$) is different from the straightness measurement coordinate system ($x^\prime - o - y^\prime$). As is shown in Fig. 3, there is a rotation angle of 45° between the two coordinate systems.

According to the relationship between the two coordinate systems, the transition from the position deviation of the laser spot detected by PSD in the $x - o - y$ coordinate system to straightness measurement values in the $x^\prime - o - y^\prime$ coordinate system can be derived by

In the proposed optical configuration, the reflector is a corner cube; therefore, the measured straightness deviation in an $x^\prime - o - y^\prime$ coordinate system is 2 times the actual straightness error. And the horizontal straightness error and vertical straightness error can be derived by

The straightness measurement requires that the measurement beam reflected by the HCC can be incident on the effective photosensitive area of the PSD. As is shown in Fig. 2, the measurement beam of straightness is reflected by ${{\rm PBS}_2}$, transmits the SR, and projects onto the PSD. Therefore, the photosensitive area of the PSD and the aperture diameter of ${{\rm PBS}_2}$ are the key to affecting the measurement range of straightness. Since the aperture diameter of ${{\rm PBS}_2}$ is larger than the photosensitive area of the PSD, the photosensitive area of the PSD determines the measurement range of straightness. The measurement range of straightness can be calculated by

#### B. Measurement Principle of Straightness Error’s Position

As shown in Fig. 2, the position of the straightness error is measured by using a designed heterodyne interferometer. In the interferometer, HCC is a hollow corner cube that can avoid the coupling influence of angle error on position measurement. The interference signal is processed by using our previously proposed signal-processing method, which is based on a phase shift of the reference signal [21]. This method guarantees the correct combination of integer and fraction fringe counting. The position of straightness error can be calculated by

where $N$ and $\varepsilon $ are the integer fringe counting number and fraction fringe number, respectively, $\lambda $ is the wavelength of the stabilized laser, and $n$ is the refractive index of air.## 4. DISCUSSION

As is shown in Fig. 1, the SR is the key optical element in the proposed optical configuration. In the configuration, the measurement beam is split into two beams by the SR, the reflected beam is used for generating an interference measurement signal, and the transmitted beam projecting onto the PSD is used for generating a laser spot position signal. The normal operation of the system requires that the measurement beam incident to the SR should be perpendicular to the incident surface of the SR. However, the perpendicularity error sometimes exists. Therefore, the influence of angle error of the SR on straightness measurement should be considered. As is shown in Fig. 4, when the SR has an angle error of $\gamma $, the transmitted beam incident to PSD deviates from its original optical path, which has an influence on the detected position of the laser spot.

According to the geometric relationship in Fig. 4, when the SR has an angle error of $\gamma $, the incident beam is refracted onto the incident surface of the SR, and the distance between the refraction beam and the original incident beam on the exit surface of the SR can be derived by

According to Eq. (5), the influence of the angle error of the SR on straightness measurement can be derived by

The influence of the angle error of the SR on straightness measurement has been simulated by using MATLAB; the simulation result is shown in Fig. 5. In the simulation, the parameters are specified as $h = {3}\,\,{\rm mm}$, $n = {1.000292}$, and $n^\prime = {1.51680}$, respectively.

Figure 5 indicates that when the angle error $\gamma $ of the SR increases from $ - {10}\,\,{\rm arcsec}$ to 10 arcsec, the influence of the angle error on the straightness measurement increases. The maximum deviation of straightness caused by the angle error of the SR is about 0.05 µm. Therefore, in order to realize a higher-precision straightness error measurement, the angle error of the SR should be controlled to a small value, and the influence of the angle error of the SR can be reduced by improving the alignment of the optical path.

## 5. EXPERIMENTS

To verify the feasibility of the proposed system for measuring two-dimensional straightness errors and their positions, an experimental setup was constructed, as shown in Fig. 6. In this setup, a dual-frequency He–Ne laser (5517B, Agilent) was used for emitting a pair of orthogonal beams with the frequency difference of 2.26 MHz and a wavelength of 632.991372 nm. A two-dimensional position-sensitive detector (PDP90A, Thorlabs) with a resolution of 0.75 µm was used for measuring straightness. One PIN photodetector (PT-1303C, Beijing Pretios) was used to detect the measurement signal with the maximum detection frequency of 10 MHz. One precision linear stage (XML350, Newport) with the resolution of 10 nm was used as a displacement driver for the position measurement experiment. Another electric linear stage (WN210TA, Beijing Micro-nano) was used for a straightness measurement comparison experiment. A commercial interferometer (5529A, Agilent) was used to measure straightness and displacement of the measured stage for comparison.

#### A. Stability Test

Good stability is a prerequisite of a measurement system that can be applied in actual measurement. To verify the stability of the proposed system for measuring straightness errors and their position, the stability test was performed. Within 1 h, the HCC is in a static state, and the straightness errors and their position are measured and recorded every 4 s. According to the test results shown in Fig. 7, the maximum deviations of horizontal straightness and vertical straightness error are $ - {0.51}\,\,\unicode{x00B5}{\rm m}$ and 0.6 µm, respectively, and standard deviations within 4 s are 0.07 and 0.07 µm, respectively. The maximum displacement deviation within 1 h and standard deviation within 4 s are 30.32 and 0.41 nm, respectively. The test results show that the proposed three-degrees-of-freedom measurement system has a good static stability, and it can be used for straightness and displacement measurement.

#### B. Repeatability Test

In this experiment, the XML350 linear stage was used for the measurement repeatability test. The stage was measured 3 times, and the displacement measurement increment step was 5 mm in the range of 300 mm. In each time measurement, the stage was driven to move to the 60 target measurement points with the displacement moving step of 5 mm; the two-dimensional straightness errors and the position data are measured and recorded at each measurement point. When this measurement is finished, the stage is driven to move to the beginning point and starts the second time measurement until three time measurements are finished. The experimental results are shown in Fig. 8. By using the two-endpoint straightness evaluating method, the horizontal straightnesses are 10.6, 10.97, and 10.34 µm, respectively, and the vertical straightnesses are 7.65, 7.77, and 7.70 µm, respectively. This experiment demonstrates that the proposed system can realize three-degrees-of-freedom measurement simultaneously, including two-dimensional straightness errors and their positions.

#### C. Straightness Measurement Range Test

This test was to verify the actual straightness measurement range. In the three-degrees-of-freedom measurement system, the diameter of photosensitive area of the PSD (PDP90A) used is 9 mm, and the diameter of the laser beam is 6 mm. According to Eq. (3), the theoretical positive and negative straightness measurement range are both 0.75 mm. In this test, considering the symmetry of the measurement range, the positive straightness measurement range is verified. The XML350 linear stage was fixed perpendicularly to the displacement measurement axis of the proposed system, the HCC was mounted on the moving element of the stage, and the laser beam was adjusted to project on the center of the PSD. The HCC moved with a step of 50 µm in a range of 0.75 mm along the positive horizontal direction, and the horizontal straightness was measured in each step. The measurement result is shown in Fig. 9, which shows that the actual straightness measurement range is consistent with the theoretical range. Figure 9 also shows that the maximum deviation and standard deviation between the proposed system and linear stage are 0.57 and 0.38 µm, respectively.

#### D. Straightness Measurement Comparison Experiment

This experiment was carried out to verify the feasibility of the proposed system for measuring straightness, and the Agilent interferometer was used for a straightness measurement comparison. In the experiment, the HCC and the Wollaston prism of the Agilent interferometer were mounted on the moving element of the WN210TA linear stage, and the moving element was moved with a displacement increment of 5 mm within a range of 300 mm. When the measurement begins, the two measurement systems measure the first position point of the XML350 stage and record data simultaneously. Then the stage was driven to move to the next point with the moving displacement of 1 mm, and the two measurement systems measured and recorded the position data again. By using the same measuring method, 300 position data of the stage are measured sequentially. The position measurement comparison error can be obtained by subtracting the two measured position data from the two measuring systems at the same measuring point. The experimental results shown in Fig. 10 indicate that the straightness error obtained by the proposed system is in agreement with that obtained by the Agilent interferometer. The maximum deviation and the standard deviation of measured straightness error between the proposed system and the Agilent interferometer are 1.15 and 0.35 µm, respectively. By using the two-endpoint straightness evaluating method, the straightness obtained by the two measurement systems are 15.62 and 16.44 µm, respectively. The experimental results verified the effectiveness of the proposed method in the application of straightness measurement.

#### E. Straightness Measurement Comparison Experiment

The experimental setup in this experiment is shown in Fig. 6. The HCC of the proposed system and displacement reflector of the Agilent interferometer were mounted on the moving element of the XML350 stage, and the moving element was moved with a step of 1 mm within a range of 300 mm. The displacement of the XML350 stage was measured simultaneously by the two measurement systems. The comparison experimental results are shown in Fig. 11, which indicates that the maximum displacement measurement deviation between the proposed system and the Agilent interferometer is 39.9 nm, with a standard deviation of 13.39 nm. This shows that the proposed system can be used to realize precision displacement measurement and can be used for measuring the position of straightness error in the proposed three-degrees-of-freedom measurement system.

## 6. CONCLUSION

In this paper, a three-degrees-of-freedom measurement system based on the Faraday effect is proposed. The advantage of the system is that not only can the two-dimensional straightness errors be measured, but also the position of straightness error can be provided simultaneously. In the proposed system, a two-dimensional position-sensitive detector is used to realize the horizontal and vertical straightness error measurements, and the laser heterodyne interferometer simultaneously gives the position of the measured straightness error. The measurement principles of straightness errors and their position are described in detail. The influence of the angle error of the semitransparent mirror on straightness error measurement is discussed. A series of experiments were performed to verify the feasibility of the proposed system. The stability and repeatability tests demonstrate that the proposed system can be used to realize three-degrees-of-freedom measurement simultaneously. The comparison experiments show that the proposed system can be used in the application of straightness error and its position measurement. All these indicate that the proposed method could be applied in precision measurement and calibration fields, and the optical configuration is easy to build and integrate.

## Funding

National Natural Science Foundation of China (51605445, 51375461, 51527807); Changjiang Scholar Program of Chinese Ministry of Education (IRT_17R98); China Postdoctoral Science Foundation (2016M601969); The Young Researchers Foundation of Zhejiang Provincial Top Key Academic Discipline of Mechanical Engineering of Zhejiang Sci-tech University (ZSTUME02B07).

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