A laser heterodyne interferometer with rotational error compensation is proposed for precision displacement measurement. In this interferometer, the rotational error of the measured object is obtained by using an angle detecting unit which is composed of a semi-reflective film, a polarizing beam splitter, a quarter-wave plate, a convex lens and a two-dimensional position sensitive detector. And the obtained rotational angle is used for compensating its influence on displacement measurement result. The optical configuration of the proposed interferometer is designed, and the mathematical model for displacement measurement with rotational error compensation is established. The coupling effect of axial displacement on rotational angle measurement and the rotational angle range used for compensation on displacement measurement are discussed in detail. To verify feasibility of the proposed interferometer, the experimental setup was constructed and several verification experiments were performed.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
For advantages of direct traceability to the standard of length and large measurement range with nanometer accuracy, laser interferometry is widely used in various fields of precision measurement, such as displacement measurement of stages [1, 2], precision machining [3, 4], semiconductor photolithography , gravitational wave detection  and calibration of measurement tools [7, 8].
Heterodyne interferometry can obtain sub-nanometer displacement uncertainty over a large dynamic range because of a high signal-to-noise ratio phase determination . In traditional heterodyne displacement interferometers, a plane mirror or a corner cube is usually employed as the measuring reflector . When a plane mirror is chosen as the measuring reflector, the titling error of the mirror will cause the reflected measuring beam to deflect its original incoming path. This has adverse effect on correct return of the measuring beam and normal generation of interference signal, and furthermore the tilt error influences the measurement result either. When a corner cube is chosen, the rotational error will have also serious effect on the optical path inside the corner cube although the measuring beam can return in a certain range of rotational angle. In these heterodyne displacement interferometers, the common problem is that the rotational error of measuring reflector will couple into displacement measurement result. Therefore, the coupling influence of rotational error should be determined and compensated in order to achieve higher precision displacement measurement. Some research works have been carried out in the past few years. For example, Kim et al developed an interferometric calibration system for linear dimension artefacts by using active compensation of angular motion errors to reduce the influence of Abbe error . Chen et al proposed a six degrees of freedom measurement system to eliminate the influence of the rotational errors produced by the measuring reflector on the measurement of straightness and its positions . Yu et al developed a compact six degree-of-freedom measurement system for precision linear stage metrology, in which a differential wavefront sensing technique was used to measure and decouple three degrees of freedom of displacement, pitch and yaw . Clark et al presented a methodology for position and orientation measurement of stages, in which a feedforward–feedback compound controller was established to provide counter-rotation of the measured target in order to reduce the misalignment error induced by angular motion . In view of the above researches, the rotational error is usually detected by using interferometry or position sensitive sensors to compensate its influence on accuracy of linear displacement measurement . Comparing with interferometry, the use of position sensitive sensors is relatively a simple and direct way to detect rotational errors.
In our previous works [16, 17], we proposed an orthogonal return method for linearly polarized beam and designed a laser heterodyne interferometer for simultaneously measuring displacement and angle based on the Faraday effect. Although this orthogonal return method can guarantee correct return of the measuring beam even large rotational motion of the measuring reflector occurs, the compensation of the influence of rotational error on displacement measurement is not considered in . In , the designed interferometer is complicated because of using two optical setups of the orthogonal return method and only realizes two degrees of freedom measurement. In this paper, a laser heterodyne interferometer with rotational error compensation is proposed. The optical configuration is designed based on the orthogonal return method, an angle detecting unit by using a two-dimensional position sensitive detector is designed, and the obtained rotational angle is used for compensating its influence on optical path length in displacement measurement. Comparing with traditional laser displacement interferometers without rotational error compensation, the proposed interferometer cannot only reduce the influence of rotational error on displacement measurement but also achieve three degrees of freedom measurement. The measurement principle is described in detail, the displacement measurement mathematical model with rotational error compensation is derived, and three experiments are performed to verify the feasibility of the proposed interferometer.
The configuration of the proposed interferometer is shown in Fig. 1. As shown in the blue dashed box, an angle detecting unit (ADU) is designed to determine the rotational error of the measured object. A stabilized dual-frequency He-Ne laser as light source emits an orthogonally linearly polarized beam with dual frequencies of ƒ1 and ƒ2. The laser beam is divided by a beam splitter (BS) into two beams. The reflected beam passes through an polarizer (P1) and projects onto the first photodetector (PD1), then the reference signal is produced. The transmitted beam is split by a polarizing beam splitter (PBS1) into a reference beam (RB) with the frequency of ƒ1 and a measurement beam (MB) with the frequency of ƒ2, respectively. The RB reflected by a plane mirror (RM1) passes through a quarter-wave plate (QP1) twice and transmits PBS1. The MB passes through a Faraday rotator (FR) and PBS2, and then incidents onto a corner cube prism (CC) which is coated with a semi-reflective film (SR). The MB is split by SR into two parts. The reflected part transmits QP2, and then is reflected by PBS2, after transmitting a convex lens (CL), it projects onto a two-dimensional position sensitive detector (PSD). The transmitted part is reflected by CC and passes through QP2. And it is reflected by PBS2 and incidents onto RM2 vertically, then this beam is reflected back along the incoming optical path. The returned MB and RB project onto PD2, then the measurement signal is produced. By detecting and processing of the reference and measurement signals, the measurement result of initial displacement can be obtained. By processing the laser spot position signal detected by ADU, the rotational error of MR can be obtained, including yaw and pith errors. Finally, the initial displacement and the rotational error data are transmitted to a computer for data processing, and then the compensated displacement result can be obtained by using the proposed compensation method.
Figure 2 shows the schematic for measuring displacement with rotational error compensation. When CC moves a displacement with a rotational angle of θ from the initial position P0 to P1, the measurement beam can return effectively in this novel optical design for displacement measurement. The rotational angle has an effect on the optical path length (OPL) inside and outside CC. The increment of OPL inside CC ΔLin(θ) is proportional to the incident angle and can be derived by 
When the center of CC’s incident plane is chosen as the rotatory center, according to the geometric relationship in Fig. 2, the decrement of OPL outside CC can be derived by
According to Eq. (3), the rotational angle θ must be measured beforehand in order to compensate its influence on OPL for precision displacement measurement. And the rotational angle can be obtained by processing the laser spot position signal detected by ADU. The rotational angle of θ can be derived by
Equation (4) shows that the rotational angle is mainly decided by the yaw and pitch angles of CC and is not influenced by the position of CC, which is discussed in detail in section 4. Then, substituting θ into Eq. (3), the compensated displacement can be obtained.
4.1 Influence of displacement on rotational angle measurement
When the measured object with a rotational angle of θ moves an axial displacement, the schematic of the influence of displacement on rotational angle is shown in Fig. 3. CL is used for eliminating the coupling effect of the axial displacement. When the rotational angle of the measured object is θ from position P0 to P1, the laser beams reflected by SR on CC are parallel with each other before reaching the incident surface of CL. According to the focusing principle of convex lens, because the detecting surface of PSD is fixed at the focus of CL, the focal points of the laser beams reflected by SR are at the same position on the detecting surface of PSD. Therefore, the axial displacement of ΔL has no influence on the rotational angle measurement by using CL, and the measured rotational angle θ can be obtained without coupling effect and used to compensate the influence of rotational angle on displacement measurement.
4.2 Rotational angle determining range for compensation
For the proposed interferometer, although the measurement beam can return effectively even CC has large rotational angle more than ± 10° , the maximum determining rotational angle is limited for the reason that the reflected beam could deflect out of the limiting aperture of CL. The rotational angle includes yaw and pitch errors. According to the geometric relationship in Fig. 3, when SR has clockwise yaw error (Yaw + ) and counter clockwise yaw error (Yaw-), the maximum measurement angle of yaw can be derived by
Figure 4 shows the simulation result of the influence of the position of SR on the maximum measurement angle when SR has yaw motion. The parameters are specified as S1 = 30 mm, d = 6 mm, and ΔR = 4 mm, 6.35 mm, and 8 mm, respectively. As shown in Fig. 4, the maximum measurement angle of yaw decreases from 6.10° to 0.34° when SR moves from 0 mm to 500 mm with ΔR of 6.35 mm, and the maximum measurement angle of yaw decreases from 1.76° to 0.88° with different the limiting aperture of CL at the position of SR of 100 mm.
When SR has clockwise pitch error (Pitch + ) and counter clockwise pitch error (Pitch-), the maximum measurement angle of pitch can be derived by
Figure 5 shows the simulation result of the influence of the position of SR on the maximum measurement angle when SR has pitch motion with the same specified parameters with SR’s yaw motion. Figure 5 indicates that the maximum measurement angle of pitch decreases from 5.97° to 0.34° when SR moves from 0 mm to 500 mm.
The above analyses show that the maximum measurement angle will decrease with the increase of the position of SR and the maximum measurement angle is proportional to the limiting aperture of CL at the same position. In addition, when SR has roll motion at an arbitrary position, the reflected beam has no influence on the maximum measurement angle.
To verify feasibility of the proposed interferometer for displacement measurement with rotational error compensation, an experimental setup was constructed as shown in Fig. 6. In the setup, a stabilized He-Ne laser (5517B, Keysight) was used to emit a pair of orthogonal beams with the frequency difference of 2.24 MHz and the wavelength of λ = 632.991372 nm. A Faraday rotator (Fx633-6, Opto-eletronics) with the rotational angle of 45° was used to realize polarization direction transition of 90°. A two-dimensional position sensitive detector (PDP90A, Thorlabs) was used to detect the laser spot position signal with the resolution of 0.675 μm for rotational angle measurement. One PIN photo-detector (PT-1303C, Beijing Pretios) was used to detect the measurement signal with the maximum detection frequency of 10 MHz, and the reference signal was provided from the rear of the laser head. One FPGA-based signal processing board with kernel chip EP2C20Q240I8 was designed for interferometric signal acquisition and phase measurement, and the resolution of the designed processing board is 0.89 nm for displacement measurement. A commercial interferometer (XL-80, Renishaw) was used to test displacement and rotational angle of the measured stages for comparison. Three experiments were carried out, the first experiment is rotational angle test, the second one is rotational angle compensation, and the third one is displacement measurement with rotational error compensation.
5.1 Angle measurement comparison experiment
This experiment was carried out to verify the feasibility of rotational angle measurement by using the designed ADU. In this experiment, a rotation stage (M-038.DG1, Physik Instrumente) with the design resolution of 3.5E-05° is used to generate rotational motion for angle measurement comparison. The measuring reflectors of the proposed interferometer and the XL-80 interferometer were mounted on the M-038.DG1 stage. The reflectors were moved with a rotational angle increment of 0.0001° in range of 0.008° and the two interferometers measured the rotational angle simultaneously. As shown in Fig. 7, the experimental results indicate that the maximum angle deviation between the proposed and Renishaw interferometer is 5.9E-05° with the standard deviation of 2.38E-05°, and the angular resolution is better than 0.0001°. The experiment results show that the ADU can measure rotational angle accurately and the measured rotational angle can be used for rotational error compensation in displacement measurement.
5.2 Rotational angle compensation experiment
In this experiment, the M-038.DG1 stage was fixed on a precision linear stage (XML350, Newport) with the positioning accuracy of 0.05 μm, and the M-038.DG1 can be moved to an arbitrary position by XML350. For an instance, the position was defined as the zero point of displacement where the S1 = 320 mm, S2 = 90 mm, the MR was moved with an angle increment of 0.001° in range of 0.12° to represent the rotational error. In each increment, the proposed interferometer measured the rotational angle and displacement simultaneously. The experimental results are shown in Fig. 8, and the data shown with red dot is the compensating displacement data. Figure 8 indicates that the maximum displacement deviation before compensation is 6.268 μm at the position of the largest rotational error, and the maximum displacement deviation after compensation is 11.8 nm with the standard deviation of 4.7 nm. The experimental results demonstrate that the rotational error has serious influence on displacement measurement, and the influence grows larger with the increase of the rotational angle. However, after compensation, the displacement deviation is reduced greatly. This experiment verifies the feasibility of the rotational error compensation method for precision displacement measurement by using the proposed interferometer.
5.3 Displacement measurement with rotational error compensation
In this experiment, the XML350 stage was used as the measured object to verify feasibility of displacement measurement with rotational error compensation. The proposed interferometer and the Renishaw interferometer simultaneously measured the same displacement provided by the XML350 stage with a displacement increment of 1mm. The experimental results are shown in Fig. 9. Figure 9(a) shows that the rotational error of the measured XML350 stage increases from 1.86E-05° to 9.25E-04° when the displacement increases from 0 mm to 165 mm. Figure 9(b) shows that the maximum displacement deviation between the proposed interferometer and the measured stage is 0.924 μm before compensation, and maximum displacement deviation between the Renishaw interferometer and the measured stage is 0.938 μm. The experimental results indicate that the displacement deviation obtained by the proposed interferometer without compensation is in agreement with that obtained by the Renishaw interferometer. But the deviations seriously deviate from the unidirectional repeatability of 0.05 µm given in the datasheet of the measured stage, which is caused by the rotational error of the measured stage. After compensation, the maximum displacement deviation between the proposed interferometer and the measured stage reduces to 0.192 μm with the standard deviation of 81 nm. These experimental results verify the effectiveness of the rotational error compensation method in application of precision displacement measurement.
In this paper, a laser heterodyne interferometer that is capable of compensating the influence of rotational error of the measured object on displacement measurement result is proposed. An ADU is designed for determining the rotational error of the measured object. And the rotational error compensation method is proposed for precision displacement measurement. One advantage of the interferometer is that three degrees of freedom measurement can be achieved simultaneously. The other advantage is that the rotational error can be measured without the influence of displacement and the displacement can be obtained accurately by using the proposed rotational error compensation method. Three experiments were carried out to verify the feasibility of the proposed interferometer. The angle comparison experiment shows that the proposed interferometer’s angular resolution is better than 0.0001°. In the rotational angle compensation experiment, the maximum displacement deviation is reduced from 6.268 μm to 11.8 nm by compensation. And the third experiment verifies the effectiveness of the proposed interferometer in application of precision displacement measurement. All these indicate the feasibility and effectiveness of the proposed rotational error compensation method. The ADU can be adopted in traditional heterodyne interferometer for rotational error detection and displacement compensation, and the proposed interferometer can be applied in precision displacement measurement, multiple degrees of freedom measurement and calibration fields.
National Natural Science Foundation of China (NSFC) (51605445, 51375461 and 51527807); Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R98); China Postdoctoral Science Foundation (2016M601969).
Authors acknowledge the Science Foundation of Zhejiang Sci-Tech University.
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