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A laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect is proposed. The optical configuration of the proposed interferometer is designed and the mathematic model for measuring displacement and angle is established. The influences of the translational, lateral and rotational movements of the measuring reflector on displacement and angle measurement are analyzed in detail. The experimental setup based on the proposed interferometer was constructed and a series of experiments of angle comparison and simultaneous measuring displacement and angle were performed to verify the feasibility of the proposed interferometer for precision displacement and angle measurement.
© 2014 Optical Society of America
1. Introduction
With the rapid development of nanotechnology and precision engineering, there are great demands for measurement instruments applicable to multiple degrees of freedom (DOFs) measurement with nanometer accuracy [1–4]. For example, the width of the smallest structures of very large scale integrated circuit manufacturing is now 22nm in semiconductor lithography, in order to realize the alignment of silicon wafer and mask precisely, three degrees of freedom with two lateral displacements and one rotational angle of the x-y-θ scanning stage need to be measured correctly [5, 6]; for precision CNC machine tools, the ultra-precision manufacturing accuracy has entered the era of nanometer scale, multi-axes synchronous manufacturing demands that the multi-dimension motion of the ultra-precision machine tool table must be measured and calibrated [7, 8]; for various precision linear stages providing nanometer movement in the field of nanometer scientific research, the measurement and calibration of these stages are necessary [9]. Therefore, single degree of freedom displacement or angle measurement does not meet the requirement of the above-mentioned measurement. High precision multi-degrees of freedom simultaneous measurement plays an important role in these precision engineering.
Current multi-degrees of freedom measurement methods mostly employ position sensitive detector (PSD) or quadrant photodetector (QPD) to realize the simultaneous measurement of multiple degrees of freedom of the measured object. For example, W. S. Park et al proposed a 6-DOF measuring system composed of a 3-facet mirror, a laser source and three PSDs to determine 6-DOF displacement of rigid bodies [10]. C. J. Chen et al presented an optoelectronic measurement system for measuring 6-DOF motion error of rotary parts with three 2-axis PSDs [11]. W. Jywe et al presented an optical calibration system for five-axis CNC machine tools with two two-demensional QPDs and two laser sources [7]. W. Y. Jywe et al presented a five-DOF measuring system composed of a micro-interferometer, a DVD pickup and a QPD to simultaneously measure machine tools [8]. X. H. Li et al presented a multi-axis surface encoder that can measure six-DOF translational displacement motions and angular motions of a planar motion stage by means of a planar scale grating and two QPDs [12]. Although these methods can realize large measurement range, the pixel’s size of the PSD or QPD limits the improvement of measurement accuracy. It is well known that laser interferometers can realize nanometer measurement accuracy. But laser interferometer is usually used to measure single DOF separately to achieve multiple DOFs measurement. The reason is that the interference will not formed when an angular rotation of the measured object occurs in a plane mirror interferometer or when a lateral movement of the measured object occurs in a corner cube interferometer. Hence, the simultaneous measurement of the displacement and angle is the primary problem to be solved in laser interferometric Multi-DOF simultaneous measurement.
In this paper, a laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect is proposed. Compared with the multi-DOF measurement methods based on PSD or QPD [7,10] that not only have micrometer lower measurement accuracy but also need calibration by using length standard tools, the proposed interferometer can achieve nanometer higher measurement accuracy as well as can be direct traceable to the definition of the meter. In addition, Compared with conventional laser interferometer, such as Agilent 5529A interferometer, Renishaw XL-80 interferometer, the proposed interferometer can realize the simultaneous measurement of displacement and angle. Furthermore, the advantage of the proposed interferometer is that the correct interference is guaranteed whether the measured object translates or rotates. The design and realization of the proposed interferometer is described in detail and several experiments were performed to demonstrate its feasibility.
2. Configuration
Figure 1 shows the proposed configuration of the laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect. A stabilized 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: one beam, as the first measuring beam, passes through BS and enters the first measuring path; another beam, as the second measuring beam is reflected by BS. The first measuring beam is split by a polarizing beam splitter (PBS1) into a reflected beam (RB) with the frequency of ƒ1 and a transmitted beam (TB) with the frequency of ƒ2, respectively. RB is reflected by a plane mirror (RM1), passes a quarter-wave plate (QP1) twice and then transmitted PBS1. TB perpendicular to the page plane passes through a Faraday rotator (FR1) with the rotational angle of 45°. The polarization direction of TB changes 45° and passes through PBS2 and QP2 and then incidents on to a corner cube prism (MR1) in the assembled measuring reflector (RR). The reflected beam by MR1 returns and parallels with incoming beam due to MR1’s inherent reverse character, then passes through QP2 and reflected by PBS2 and then incidents on to the fixed plane mirror (RM2) vertically, then, this beam is reflected back along the incoming optical path and passes through FR1 again. According to the Faraday effect, a linearly polarized beam passes through a Faraday rotator with a rotational angle of θ forward and backward, the polarization direction of the beam changes 2θ. Here, θ = 45°, then the polarization direction of the return beam changes 90° which is orthogonal with that of original input TB, then the return beam is reflect by PBS1. RB from reference arm and TB from measuring arm recombine into one beam which passes through the first polarizer (P1) and projects onto the first photodetector (D1) to generate the first measurement signal. In the first measuring path, PBS2 and RM2 are placed at a rotational angle of 45° with respect to the page plane, and the fast axis of QP2 is parallel to the page plane. The second measuring beam is reflected by a mirror (R) and enters the second measuring path which has the same optical layout as the first measuring path and finally generates the second measurement signal. The reference signal is provided from the rear of the laser. The displacement and rotational angle of RR can be obtained by acquisition and processing of the three interference signals.
Fig. 1 Schematic of laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect.
In this optical configuration, FR is used to rotate the polarization direction of a linearly polarized beam rather than using a quarter-wave plate or a half-wave plate. The reason is as follows: If a quarter-wave plate is used, when the polarization direction of a linearly polarized beam is 45° with respect to the fast axis of QP, the beam passes through QP twice forward and backward, the polarization direction of the returned beam can be orthogonal to that of the original input beam. But, the emergent light from QP first time is a circularly polarized beam. This circularly polarized beam does not satisfy the demand of a linearly polarized beam that projects onto PBS2 (or PBS4) in the proposed interferometer. If a half-wave plate is used to rotate the polarization direction of a linearly polarized beam, no matter what angle the input beam projects onto the half-wave plate, the polarization direction of the returned beam from measuring path will be always same as the initial polarization direction of the original input beam when a linearly polarized beam passes through the half-wave plate twice forward and backward.
3. Principle
As shown in Fig. 2, when RR moves a displacement along z axis with a rotational angle of θ around the center O from initial position P0 to P1, two parallel measuring beams project on to the corresponding MR1 and MR2. MR1 and MR2 have the same size and characteristic. The increment of the optical path length (OPL) inside MR is proportional to the incident angle and can be derived by [13]
where, θ is the incident angle, H is the height of MR, n is the refractive index of air, and n′ is the refractive index of MR’s material.According to the geometric relationship in Fig. 2, when RR rotates the angle of θ, the optical path difference (OPD) of the first measuring path (Path1) can be derived by
where, nl is the OPD at P1 without rotational angle, 2nΔL1 is the decrement of OPL outside the MR.Similarly, the OPD of the second measuring beam (Path2) can be derived by
where, 2nΔL2 is the increment of OPL outside the MR.According to Eq. (2) and Eq. (3), considering that the optical fold factor of the configuration is 4, the measured displacement and the rotational angle can be derived respectively by
Equation (4) shows that the measured displacement is not only decided by the two optical path’s OPDs, but also is influenced by the rotational angle of θ. Equation (5) shows that the angle of θ can be directly calculated by using the two optical path’s OPDs. Substituting the θ into Eq. (4), the measured displacement can be obtained.
4. Analysis and discussion
4.1 Translational movement
During displacement and angle measurement process, RR moves with the measured object along the moving axis. The movement of RR includes two cases: one is that RR moves without rotational angle as shown in Fig. 3(a), the other is that RR moves with rotational angle as shown in Fig. 3(b). When RR moves a displacement of ΔL from P0 to P1, the OPLs inside of MRs of the two measuring optical path do not change and the increment of OPLs outside of MRs are the same which are equal to ΔL in despite of the movement with a rotational angle or not. Hence, the translational movement does not influence angle measurement; the displacement can be obtained correctly by the change of OPL in the measuring path.
4.2 Lateral movement
Figure 4 shows when RR has displacement perpendicular to the moving axis during measurement process. Figure 4(a) shows that the reflected measuring beam moves a displacement of 2Δh and the OPD in each measuring path does not change when RR has a lateral displacement of Δh perpendicular to the moving axis without rotational angle. Figure 4(b) shows the case that RR has a lateral displacement of Δh perpendicular to the moving axis with rotational angle of θ. According to the geometric relationship, in each measuring optical path, the increment OPL of incident beam and the decrement OPL of the reflected beam are equal to Δhtanθ while they have opposite sign. Therefore, the lateral movement of RR does not influence the OPL of each measuring optical path, and then will not influence the measured displacement and angle.
4.3 Rotational movement
When the rotating center of RR is at the midpoint (O) of the connected line between the two MRs as shown in Fig. 2, the displacement and angle can be calculated with Eq. (4) and Eq. (5). But, during actual measurement process, the rotating center of RR might vary with the movement of the measured object. This will influence the measurement. The influence is analyzed and discussed as follows.
Figure 5 shows that when RR rotates an angle of θ around the rotating center of O′ instead of O from position P0 to P1, the process of RR’s movement can be divided into two steps: first, RR rotates a angle of θ around O and then moves a lateral displacement along axis y; second, RR moves a translational displacement along the axis z. According to above analysis, the first step only changes the rotational angle and does not influence the displacement measurement. According to the geometric relationship in Fig. 5 the second step changes the displacement of d[sin(α + θ)-sinα] while has no influence on the rotational angle. Hence, when the rotating center of RR changes, it does not influence the measurement of angle, but will influence the displacement measurement. In this case, Eq. (4) should be modified by
where, d is the distance from O to O′, α is the angle between the connected line of two rotating centers and axis y.Equation (6) shows that in order to realize correct displacement measurement, the position relationship between the center O of RR and the center O′ of the measured object should be known beforehand. In actual measurement, for convenience, it is better to let the centers of O and O′ be aligned on the same line along axis z as shown in Fig. 6. Then Eq. (6) is given by
4.4 Discussion of influence of the rotational angle error of Faraday rotator
The proposed interferometer is based on heterodyne interferometry and it also involves nonlinear periodic error problems. Because the nonlinear periodic errors about heterodyne interferometer have been discussed in many other research works [14–18], the same problems about these nonlinear errors are not discussed here. In the proposed interferometer, the Faraday rotator is a crucial optical device different from other heterodyne interferometers and its influence is discussed as below.
When the rotational angle of FR is not 45° with a rotational angle error of α, the return TB with the frequency of f2 will not be orthogonal to the return RB with the frequency of f1 as shown in Fig. 7. The y component of the return TB and the return RB generate the heterodyne measurement signal.
The electric fields of the y component of the return TB and the return RB can be expressed as
where E1 and E2 are the amplitudes, φ1 and φ2 are the initial phases.Then the measurement signal is expressed as
where Δφ = φ1-φ2 is the phase difference, Δf = f1 - f2 is the frequency difference.According to Eq. (9), a simulation to determine the influence of the rotational angle error of FR on the measurement signal is shown in Fig. 8. The curve with α = 0° represents the case without rotational angle error. When FR has rotational angle errors of α = 1°, 3° and 5°, the amplitude of the interference signal will decrease gradually, but the zero phase does not change. Thus, the rotational angle error of FR does not induce extra nonlinear error in the proposed interferometer.
From Fig. 7, the x component of the return TB will pass PBS1 (or PBS3) and enter the laser source, this will cause unstable output of laser beam. So, in actual application, a precision rotational angle of FR should be chosen to avoid the influence of feedback beam on laser source.
5. Experiments
To verify the feasibility of the laser heterodyne interferometer for measuring displacement and angle based on the Faraday effect. An experimental setup was constructed as shown in Fig. 9. In this setup, the laser source is a dual-frequency stabilized He-Ne laser (5517B, Agilent Co., USA) which emits a pair of beams with the frequency difference of 1.9 ~2.4 MHz and the wavelength of λ = 632.991372nm. Two Faraday rotators (Fx633-6, Opto-eletronics Co., China) were used with the rotational angle of 45° ± 0.5°. Two high-speed PIN photodetectors (PT-1303C, Beijing Pretios Co., China) were used to detect the two measurement signals with the maximum detection frequency of 10MHz. The reference signal is provided from the rear of the laser. Signal processing board is developed with the dual-mode phase measuring method proposed previously [19], its displacement resolution is better than 0.03nm and the center distance of two MRs in RR is 150mm. Two experiments were carried out, one is the comparison experiment for measuring angle with a commercial interferometer, and the other experiment is simultaneous measurement of displacement and angle of two precision stages.
5.1 Angle comparison experiment
In this experiment, the proposed interferometer and a commercial interferometer (XL-80, Renishaw Co., UK) tested the same angle of a rotation stage (M-038.DG1, Physik Instrumente Co., Germany) for comparison. The resolution of the rotation stage is 3.5 × 10−5 ° (0.60 μrad). The measuring reflectors of the two interferometers were fixed on the rotation stage.
The first experiment is the angle measurement test with the step of 1° in the range of ± 10°. When the rotation stage moved an angle step of 1°, the proposed interferometer and Renishaw interferometer detected the increment angle simultaneously. The experimental result is shown in Fig. 10. The deviation is the difference between the measured angles of two interferometers. Figure 10 indicates that the average angle error is −0.2 × 10−5 ° and the standard deviation is 0.9 × 10−5 °.
The second experiment is the angle measurement test with the step of 0.0001° in the range of 0.105°. The experimental result is shown in Fig. 11. The experimental result indicates that the average angle error is 0.1 × 10−7 ° and the standard deviation is 0.54 × 10−5 °.
The above angular experimental data show that the measuring results of the proposed interferometer are coincident with those of the Renishaw interferometer. This demonstrates that the proposed interferometer can realize precision angle measurement.
5.2 Displacement and angle measurement experiment
To verify the proposed interferometer for simultaneous measuring displacement and angle, a precision linear stage (M-521.DD, Physik Instrumente Co., Germany) with the displacement resolution of 0.01µm and the pitch/yaw of ± 35µrad and a nanopositioning stage (P-752.11C, Physik Instrumente Co., Germany) with the displacement resolution of 0.1nm and the pitch/yaw of ± 1µrad were measured with the proposed interferometer, respectively. Before measurement began, RR’s center was aligned along the moving axis of a stage.
Firstly, RR fixed on the table of the linear stage was moved with the step of 1mm, the displacement and yaw angle of the linear stage were determined simultaneously. The experimental results are shown is Fig. 12. Figure 12(a) shows that the average displacement error is −0.003µm and the standard deviation is 0.693µm and Fig. 12(b) shows the yaw angle of the measured linear stage is 0.00184° (32.11µrad).
Secondly, the nanopositioning stage moved RR with the step of 50nm. The proposed interferometer simultaneously measured the displacement and yaw angle. Figure 13 shows the experimental results of nanometer scale measurement. Figure 13(a) shows that the average displacement error is −1.66nm and the standard deviation is 2.4nm and Fig. 13(b) shows the yaw angle of the measured nanopositioning stage is 0.9 × 10−5 ° (0.16µrad).
The above experimental data indicate that the measuring results of the proposed interferometer are coincident with those of the linear stage and the nanopositioning stage. These experimental results demonstrate that the proposed interferometer can be used to realize precision displacement and angle simultaneous measurement.
6. Conclusion
In this paper, a laser heterodyne interferometer that is capable of simultaneous measuring displacement and angle based on the Faraday effect is proposed. The configuration and principle of this interferometer are described in detail. And the influences of different movement of the measured object on displacement and angle measurement are analyzed. One advantage of this interferometer is that the correct interference is guaranteed whether the measured object translates or rotates. Another advantage of this interferometer is that not only the displacement of the measured object can be determined, but also the angle of the measured object can be measured simultaneously to compensate the measured displacement. Two verification experiments were carried out based on the constructed setup of this interferometer. The angle comparison experiments of different angle ranges with a commercial interferometer were done and the experimental results show the feasibility of precision angle measurement of the proposed interferometer. The displacement and angle measurement experiments of a precision linear stage with a millimeter range and a nanopositioning stage with a micrometer range verified the capability of simultaneous measuring displacement and angle of the proposed interferometer. All these indicate that the proposed interferometer could be applied in precision measurement and calibration fields. And the multiple degrees of freedom measurement is easy to construct and realize by combining different quantities of this interferometer.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (NSFC) under grants No.51375461, No.51205365 No.51327005 and No.90923026 and the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) under grant No. IRT13097.
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