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Abstract
A novel encoder with dual displacement resolution is developed by integrating a multiple-grating-scale holographic displacement sensor and a heterodyne interferometer. With suitable arrangement of the measurement system, two effective grating pitches (0.41 μm and 10.62 μm) can be obtained and the theoretical sensitivities of them are 0.9 °/nm and 0.036 °/nm. Meanwhile, the best resolution of the proposed method can be estimated of 0.3 pm and 7.4 pm, respectively. Furthermore, displacement errors of the proposed method can be better than 0.2% for 1 mm displacement measurement. The experimental results showed that the proposed encoder provided high sensitivity, high resolution, and well against environmental disturbance.
© 2017 Optical Society of America
1. Introduction
To achieve high resolution positioning, the displacement measuring methods [1–11] become more important and have attracted great attention over the past few decades, especially applied these techniques in nano-positioning systems such as semiconductor manufacture facilities, nano-handling, nano-manipulating and nanofabricating equipment. Gao et al. [1] have reviewed the state-of-art sensor technologies for precision positioning and pointed out the advantages of the laser grating interferometer and optical encoder played an important role in ultra-precision positioning system.
Laser grating interferometry is one of the most efficient methods of the displacement measurement. This method provides better immunity against environmental disturbances [2] and has attracted more attention in in-plane displacement measurement. Hsieh and Chen [3] integrated Wollaston prism and linear grating into a novel apparatus. To analyze the interference signal of the overlapping area of the ± 1 order diffraction beams pass through the apparatus, the displacement of linear grating can be obtained. Their method provided a theoretical resolution and sensitivity of 56 pm and 0.18 °/nm, respectively. Lee and Jiang [4] developed wavelength-modulated-phase-shifting grating interferometer for in-plane displacement measurement. The results showed that the resolution and sensitivity can be estimated of 2 nm and 0.72 °/nm, respectively. Wu et al. [5] constructed a common-path laser planar encoder for measuring displacement with integrating a Fresnel zone plate and 2D grating. Their method exhibited high environmental immunity and showed that the system stability was better than 1 nm/h. Furthermore, the resolution of their system was approximated of 0.07 nm. Kao et al. [6] described a diffractive laser encoder which the apparatus is constructed with the Littrow configuration. The advantages of the Littrow structure mean that a small displacement error and high repeatability can be achieved. Their encoder provided a measurement error and repeatability approximated of 53 nm and ± 20 nm, respectively. Agarwal and Shakher [7] applied the Talbot effect of a pair of circular gratings to measure the in-plane displacement. To analyze the position shifting of the interferometric pattern of the zeroth, first, second, and third order fringes, the relative in-plane displacement can be obtained. Their results showed that the measurement errors are ± 20 μm within the displacement range of 1 mm. Otsuka et al. [8] combined two gratings, in which one of them consisted 4 grating scales. To analyze the interference signal and with the numerical interpolation technique, the displacement resolution can be achieved 5 nm and preserved 0.1 μm accuracy over 20 mm displacement. Kimura et al. [9] developed single optical encoder to measure in-plane and out-plane displacements. Their results indicated that the resolution was better than 0.5 nm and pointed out the dominated error of their sensor was resulted from the crosstalk error of the misalignment of the coordinate system. Li et al. [10] improved Kimura’s optical encoder and integrated with a novel coding system for absolute displacement measurement. Their results showed that the accuracy could be achieved of 0.5 μm and the displacement error in x-direction would be better than 0.6% as the displacement range within 4 mm.
In such measurement methods described above, the grating scale is a critical issue, which affected to the sensitivity and resolution of the measurement. Small grating scale provides high sensitivity; therefore, to achieve nano- or subnano- sensitivity and resolution of the displacement measurement, the grating scale needs to be on a micron scale or lower [11]. Besides, the measurement speed of the displacement also relates to the grating scale. Smaller grating scale exhibits higher sensitivity and resolution but slower of the measurement speed (need more time to progress the phase unwrapping procedure). In contrast, larger grating scale provides higher measurement speed but lower of the sensitivity and resolution. Unfortunately, it is difficult to preserve high measurement speed, high sensitivity, and high resolution by using single grating-type sensor. In this study, we proposed a novel encoder for displacement measurement which integrated a holographic displacement sensor (HDS) into a heterodyne interferometer. The holographic displacement sensor fabricated by a holographic plate which exposure two grating scales at well-designed position on the holographic plate. Based on the arrangement of the measurement system, the HDS provides two effective grating pitches and the sensitivity can be reached 900 (°/μm) for effective grating pitch (EGP) of 0.41 μm and 36 (°/μm) for EGP of 10.62 μm. Theoretical prediction of the displacement error of the proposed method can be reached of 0.11 nm and 2.78 nm, respectively. Therefore, the proposed method can preserve high sensitivity and high resolution by using single displacement sensor. Besides, the proposed method exhibits better immunity against environmental disturbance. The stability evaluation of the proposed encoder can be better than 0.6° of phase variation within 3 min. The error analysis indicates that the displacement errors of the proposed method can be better than 0.52 nm (0.13%) and 21 nm (0.2%), respectively. Furthermore, the smallest displacement variation can be observed of 1 pm approximately. Based on these findings, the proposed encoder provide two effective grating pitches for displacement measurement by using single holographic grating, in which the proposed encoder can be integrated into a motorized stage with a suitable controlled algorithm and provide high sensitivity, high resolution, and accomplished a rapid displacement measurement.
2. Principle
The schematic diagram of this method is shown in Fig. 1. For convenience, the + z axis is chosen to be along the direction of propagation and x- axis is along the horizontal direction. A heterodyne light source [11] is incident onto grating G1 of the holographic displacement sensor (HDS). The HDS is a single holographic plate with three different exposure locations (indicated G1, G2, and G3 in Fig. 1) and various grating scales, in which the grating scales of those gratings are dg1, dg2, and dg3, respectively. In this study, we controlled the grating scales of dg2 is equivalent of dg3.
Fig. 1 Schematics of the dual displacement resolution encoder. (a) Schematic diagram; (b) photograph of the proposed encoder.
The test beam is guided into ± 1st and ± 2nd order diffracted lights and those diffracted lights will be propagated into four paths: (1) G1 → mirror M1 → mirror M2 → polarization beam splitter PBS1 → analyzers AN1 (45°) → detectors D1, (2) G1 → mirror M1 → polarization beam splitter PBS1 → analyzers AN2 (45°) → detectors D2, (3) G1 → mirror M1 → G2 → mirror M3 → polarization beam splitter PBS2 → analyzers AN3 (45°) → detectors D3, (4) G1 → mirror M1 → G3 → polarization beam splitter PBS2 → analyzers AN4 (45°) → detectors D4. According to the arrangement of the optical configuration, the interference signal detected by the detector D1 was interfered with p- polarization of the + 1st order diffracted light and s- polarization of the –1st order diffracted light. After Jones matrix calculating [11, 12], this signal can be written as
The interference signal detected by the detector D2 was interfered with s- polarization of the + 1st order diffracted light and p- polarization of the –1st order diffracted light; it can be written as
Similarly, the interference signals detected by D3 and D4 can be written as followed:
where ω is the angular frequency difference between the p- and s- polarizations of the heterodyne light source, ϕGi (i = 1, 2, and 3) is the phase shifting coming from moving grating which is dependent on the diffraction order m, grating scale dgi (i = 1, 2, and 3) and displacement S. ϕGi can be written as [11, 13-14]To compare I1 and I2 with phase meter (or lock-in amplifier), the total phase shifting Φ12 can be measured immediately. It is obvious that Φ12 = 2ϕG1-(−2ϕG1) is 4 times magnification of the phase shifting ϕG1 and measured the displacement (SEGP1) will be related to the effective grating pitch (EGP) dEGP1. Similarly, the total phase shifting Φ34 = −4ϕG1 + 2ϕG2 + 2ϕG3 can be obtained and the displacement (SEGP2) will be related to the EGP dEGP2. Hence, the displacement obtained from Φ12 and Φ34 can be expressed as:
where dEGP1 and dEGP2 can be written asAccording to Eqs. (6) and (7), the phase variation versus EGP under various combinations of the grating scales of dg1, dg2, and dg3 can be simulated and showed in Fig. 2. Figure 2(a) indicated that dEGP1 decreased as the grating scale dg1 decreased. Meanwhile, Fig. 2(b) showed that the various EGP could be obtained by the combination of dg1, dg2, and dg3. When dg2 is equivalent of dg3 in the EGP combination, dEGP2 decreased as the grating scale dg1 decreased. When dg1 is fixed at 1.6 μm, dEGP2 increased as the grating pitch dg2 (or dg3) decreased. Besides, the smallest EGP can be obtained as the combination of grating scales dg1, dg2, and dg3 are 1 μm, 0.9 μm, and 0.9 μm (EGP = 0.563 μm).
Fig. 2 Theoretical simulation of the EGP of various combination of grating pitches. (a) dEGP1; (b) dEGP2.
Furthermore, the theoretical sensitivity and displacement error of the proposed method depended on the EGP value that can be simulated and showed in Fig. 3. The sensitivity indicated by the slope of the phase-displacement curve. More precipitous of the curve, more sensitivity can be achieved. Figure 3(a) showed that the sensitivity of the proposed method increased as the EGP decreased and the best sensitivity can be reached to 900 (°/μm). Figure 3(b) indicated that the theoretical displacement error of the proposed method, in which the displacement errors can be as small as 2.78 nm and 0.11 nm when measured by the EGPS of 10 μm and 0.4 μm under the phase error of 0.1°. It is obvious that the displacement error of the proposed method can be within dozens of nanometers as the phase error of 1°. Based on these simulation results, the proposed method can provide dual displacement sensitivity and preserve acceptable displacement error by using single displacement sensor.
Fig. 3 Theoretical simulation of the sensitivity and displacement error under various EGP. (a) sensitivity; (b) displacement error.
3. Experiments and results
A heterodyne light source, consisting of a linearly polarized He-Ne laser at 632.8 nm with frequency stabilized control and an electro-optic modulator EOM (model: 4001, New Focus, Inc.) driven by a function generator with 1k Hz sawtooth signal as shown in Fig. 1. The fast axis of EOM was located at 45° to the x-axis and the frequency difference between the p- and s- polarizations was 1k Hz. The test signals were received by the detectors and sent into the high speed, multichannel data acquisition card (model: USB-6343, National Instruments Corporation). Then the phase difference was measured by pc-based software, which was programmed by the Labview software (version: 7.0, National Instruments Corporation) with the dual phase-lock technology similar to the algorithm of commercial lock-in amplifier (Stanford Research Systems, SR 850) [15, 16]. The phase resolution of this program can be adjusted manually from 0.1° to smaller than 0.001°. The commercial precision laser dimensional measuring system was adopted for comparison (model: HP 5529A, Hewlett-Packard Development Company) which were calibrated by the primer standard system in National Measurement Laboratory (NML) of the Center for Measurement Standards (CMS). The sampling-rate of these experiments was controlled during the whole measurement process and the value was fixed at 100 measured points per second. The HDS was mounted on the motorized stage (model: SFS-H60X and HPS60-20X-M5, Sigma Koki) and the moving speed of the stage was varied from 1 μm/s to 500 μm/s which depended on the displacement range. The maximum speed of the motorized stage was controlled at 500 μm/s for matching the acquisition rate of the self-developed program.
HDS was fabricated by recording the interference signal on a Slavich photographic plate (PFG 01) with an optimal exposure of 30 μJ/cm2. Following exposure, processing is carried out with the technique presented by Chen et al. [17] using a SM-6 developer and PBU-Amidol bleach. The HDS consisted of three diffracted gratings, in which the grating scale of G1 is approximated of 1.630 μm and the grating scales of G2 and G3 are approximated of 0.847 μm. Because of mismatching between second-order diffracted angle of G1 and first-order diffracted angle of G2 (G3), the emergent diffracted beams will be deflected to normal direction of the HDS. The physical arrangement and photograph of HDS were showed in Fig. 4. It is obvious that the measurement range of the proposed method will be limited by the exposure range of the photographic plate and the displacement measurement range of the proposed method would not be exceed 1 cm.
3.1 Effective grating pitch (EGP) measurement
The EGP can be determined by the 2π phase discontinuity of the phase-displacement curve. The displacement of the grating can be measured by HP interferometer and meanwhile the phase variation measured by the data acquisition card with self-developed lock-in amplifier program. Figure 5 showed the measurement results of EGPs dEGP1 and dEGP2, in which EGPs were 0.41 μm and 10.62 μm respectively. Based on Eq. (7), the theoretical values of EGP1 and EGP2 are approximated of 0.407 μm and 10.786 μm. In contrast, there are no significant difference between the experiment results and theoretical prediction.
3.2 Stability measurements
Figure 6 demonstrated the stability evaluation of the proposed system, in which the results indicated the phase noise level coming from the unexpected electronic noise and environmental disturbance. When the measurement progressed, the stage was held on stationary and the environmental temperature was fixed at 25 ± 1 °C. To prevent air disturbance, the windshield was adopted to use. In stationary condition, the measured phase variations are contributed from high and low frequency phase noises. Figure 6(a) shows the phase noise of the proposed system and the results indicated that the phase variation increased as the EGP increased. The phase variation of EGP2 is approximated 6 times in magnitude of EGP1. To convert the phase variation into displacement variation of EGP1 and EGP2 (showed in Fig. 6(b)), the displacement variation of EGP1 shows high stability, otherwise, the displacement variation of EGP2 is approximated of 16 nm within 3 min. We also monitor the displacement variation of the comparison method (HP interferometer, HPI) within the same observation period, the variation is approximate of 5 nm and that is worse than the results measuring by EGP1. The displacement fluctuation of HPI was 100 times larger than the result of EGP1 and 3 times smaller than the result of EGP2. Therefore, the results indicated that the environment noise immunity of the EGP1 were better than HPI. Furthermore, the smallest displacement variation can be measured by EGP1 and EGP2 which indicated at black square region in Fig. 6(b) and enlarged this region in Fig. 6(c). It is obvious that the smallest displacement variations were approximated of 1 pm (EGP1) and 0.05 nm (EGP2), respectively.
Fig. 6 Stability comparison between the proposed encoder and HPI. (a) phase variation of the proposed encoder under various EGP; (b) displacement variation comparison; (c) enlargement of the black square region in (b).
3.3 Long/short travel range measurements
To demonstrate the performance of proposed method, HDS had mounted on the motorized stage for measuring long/short travel range. Figure 7 shows the long travel range measurement results, in which the displacements were approximated of 10 μm, 100 μm (0.1 mm) and 1000 μm (1 mm). It is obvious that the displacement measured by EGP1 preserved better results than those measured by EGP2. Either the travel range approach to 1000 μm, the displacement difference between the proposed encoder and HPI were within 0.2% and 0.5% measured by EGP1 and EGP2, respectively.
Figure 8 demonstrated the short travel range measurement results, in which three waveform motions were executed, including step, sinusoidal, and triangular waveforms. The motorized stage was operated in a closed-loop function to enable forward and backward motion with a travel range within 100 nm. It is obvious that the measurement curves exhibited similar behavior between the proposed encoder and HPI. The displacement differences between EGP2 and HPI were within 3 nm as the travel range of 65 nm of the step-type movement. In contrast, the displacement difference between the proposed encoder and HPI as the travel range of 15 nm were within 1 nm and showed no significant difference between the proposed encoder and HPI as the triangular-type and sinusoidal-type movements.
Fig. 8 Short travel range measurement results. (a) 65 nm step type movement; (b) 15 nm step type movement; (c) 15 nm sinusoidal type movement; (d) 15 nm triangular type movement.
4. Discussions
The difference between the proposed method and comparison method were resulted from the displacement error of the proposed method and it can be derived from Eq. (6) and expressed as
where ∣ΔSEGP1∣ and ∣ΔSEGP2∣ indicate the displacement errors of proposed method; ∣ΔΦ1∣ and ∣ΔΦ2∣ are the phase errors resulted from the measurements by using EGP1 and EGP2; ∣ΔdEGP1∣ and ∣ΔdEGP2∣ are grating pitch errors. In this method, the phase errors (∣ΔΦ1∣ and ∣ΔΦ2∣) might due to wavelength (or frequency) shift of a laser [18], unexpected electronic variation, and nonlinearity periodic error [19]; the grating pitch error (∣ΔdEGP1∣ and ∣ΔdEGP2∣) might result from the thermal expansion of the holographic material [20], uniformity of the grating pitch [21], and the effective grating pitch judgment error.The cosine error, the misaligned angle between the grating moving axis and the measurement direction of HPI, can be controlled with mounting the grating onto a precision rotational stage. In our method, the resolution of the precision rotational stage can be reached to 0.08° and the cosine error is approximated of 0.1 nm for the travel range of 100 μm. If the misaligned angle approaches to 1°, the cosine error will be approximated of 20 nm. Therefore, the cosine error coming from the proposed method can be limited at lower level. The phase error and grating pitch error will discuss in following sections.
4.1 Phase error analysis
The wavelength shift of a stabilized He-Ne laser is about 3 × 10−4 nm and the phase error resulting from the wavelength shift can be estimated to be 9.14 × 10−5 °. Therefore, the phase error results from the wavelength shift of the stabilized He-Ne Laser can be ignored in this method. The unexpected electronic variation can be indicated by stability of the proposed system and the noise levels are approximated of 0.1° and 0.58° for measuring by EGP1 and EGP2, respectively.
The nonlinearity periodic error includes the second-harmonic error and polarization-mixing error coming from polarization components (e.g. PBS, polarizer, and grating) of the proposed system [22]. Based on Wu’s analysis [19], the residual nonlinearity is less than 0.0003°. Due to the previous evaluation [23], the nonlinearity periodic error can be simulated and showed in Fig. 9. The average value of the nonlinearity periodic error of the proposed system is approximated of 0.04° under precision conditions.
Another possible source of the phase error is coming from the A/D converter with self-developed phase-lock software (National Instrument, LabView) for measuring the phase variation. To consider the external reference phase noise and phase drift of the software, the phase error of the self-developed lock-in program is approximated of 0.05° [15, 16, 24]. That can be improved by increasing the time constant and acquisition time of program but these improvements will achieve at a cost of processing time increasing.
4.2 Grating pitch error analysis
The thermal expansion of the holographic material will affect the accuracy of displacement measurement. Holographic material (Slavich, PFG-01) was adopted to record the interference signal and became a displacement sensor. Based on previous evaluation [20], the expansion coefficient of PFG-01 is approximated of 1.55 × 10−5 μm/°C. Assuming that the temperature fluctuation is within 1 °C, the grating pitch error is approximately 0.016 nm.
The uniformity of the holographic grating can be evaluated by effective grating pitch measurement described in section 3.1 and the optical setup showed in Fig. 10(a). To evaluate uniformity of the different gratings G1, G2 (or G3), we measured 5 periods of G1 and G2 (or G3) which showed in Figs. 10(b) and 10(c). Obviously, the tendency of 2π phase discontinuity of G1 and G2 preserved high uniformity. The average value and standard deviations of grating pitch of G1 are 1.630 μm and 5.5 nm; similarly, those values of G2 are 0.847 μm and 5.5 nm. Therefore, the grating pitch error coming from the uniformity of grating might be concerned of the standard deviation of the grating pitch and approximated of 5.5 nm in this method.
Fig. 10 Uniformity of holographic grating. (a) Optical configuration; (b) results of G1; (c) results of G2 (or G3).
The effective grating pitch judgment error is resulted from the imprecise judgment of the effective grating pitch which depended on the phase resolution and phase data acquisition rate of self-developed lock-in software, and moving speed of the motorized stage. This error will become a noticeable error of the long travel range displacement measurement, especially of displacement measurement by using EGP2. To compensate this error, the accurate value of EGP2 can be determined by 2π phase discontinuity judgment of EGP1 and the results showed in Fig. 11. In this method, the phase resolution and data acquisition rate were controlled at 0.001° and 100 Hz; the moving speed of stage was controlled at 2 μm/s. It is obvious that the EGP1 can provide more precise result and corrected dEGP2 from 10.78 μm to 10.62 μm (showed in Fig. 11(b)). Based on the compensation of EGP2, the effective grating pitch error of EGP2 can be ignored of the proposed method.
Fig. 11 Compensation of the effective grating pitch error. (a) full-scale investigation of EGP1 and EGP2; (b) enlargement of black square region in (a).
According to the error analysis described above, we can summarize the values of phase error and grating pitch error of the proposed method in Table 1. To substitute these values into Eq. (8), the displacement errors (∣ΔSEGP1∣ and ∣ΔSEGP2∣) of the proposed method can be simulated and showed in Fig. 12. Figure 12(a) indicated the displacement error of EGP1 and the largest error is approximated of 0.52 nm as the motorized stage movement of 0.41 μm (measured phase of 360°) and the grating pitch error of 6 nm. Similarly, Fig. 12(b) indicated the displacement error of EGP2 and the largest error is approximated of 21 nm as the motorized stage movement of 10.62 μm and the grating pitch error of 6 nm. The error analysis can explain the difference between the proposed method and the comparison method, especially in long travel range displacement measurement.
Table 1. Error source of the proposed method
4.3 Resolution
The resolution of the displacement can be defined as the displacement changed which resulted from the minimum detectable phase of the self-developed lock-in software and can be described as
where REGP1 and REGP2 are the resolutions of the proposed method under different effective grating pitch, ΔϕEGP1 and ΔϕEGP2 are the minimum detectable phases of the proposed method. Therefore, the resolution of the proposed method can be simulated with Eq. (9) and showed in Fig. 13. If take the phase error of self-developed lock-in program as the minimum detectable phase, the resolution of the proposed method under two EGPs are 0.01 nm and 0.18 nm, respectively. In worst case, take consider of the phase errors ∣ΔΦ1∣ and ∣ΔΦ2∣ as the minimum detectable phase, the resolution of the proposed method under two EGPs are 0.06 nm and 5 nm, respectively. The best resolution of the proposed method can be obtained by considering the minimum detectable phase of self-developed lock-in program (0.001°), in which the resolution can be reached to 0.3 pm and 7.4 pm under two EGPs.5. Conclusion
A novel encoder for precision displacement measurement was developed, in which the encoder fabricated by a holographic plate and exposure two different grating scales at three well-designed positions on the holographic plate. To integrate the encoder into a heterodyne interferometer, two effective grating pitches (EGP1: 0.41 μm and EGP2: 10.62 μm) can be obtained for the displacement measurement. Based on the arrangement of the measurement system, the proposed method realized dual displacement resolution by using single encoder and therefore preserved high sensitivity and high resolution for the displacement measurement. For the consideration of the rapid precision positioning, we will develop suitable controlled algorithm by analyzing interference signal coming from EGP2 for a coarse position adjustment of the motorized stage and combined with the interference signal coming from EGP1 for fine position adjustment of the motorized stage.
Besides, the results indicated that the theoretical sensitivity and resolution can be reached 0.9 °/nm and 0.3 pm as the measurement by analyzing the interference signal coming from the effective grating pitch of 0.41 μm. Meanwhile, the proposed encoder preserved lower displacement measurement error and the error of the proposed system can be better than ± 0.5%. Experiment results also showed that the proposed system exhibited high stability and well against the environmental disturbance. The smallest displacement variation can be observed and approximated of 1 pm. Based on these findings, we can conclude that the proposed method can be applied to use as a displacement encoder for precision displacement measurement.
Funding
National Science Council of the Republic of Taiwan, China (MOST 105-2221-E-155-025).
Acknowledgments
The authors would like to thank Mr. E. William Thornton for proofreading the article.
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