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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 16 — Aug. 12, 2013
  • pp: 18872–18883
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Common-path laser planar encoder

Chyan-Chyi Wu, Yan-Zou Chen, and Chia-Huang Liao  »View Author Affiliations


Optics Express, Vol. 21, Issue 16, pp. 18872-18883 (2013)
http://dx.doi.org/10.1364/OE.21.018872


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Abstract

This paper presents a common-path laser planar encoder (CLPE) for displacement measurements in the X - and Y - axes. The CLPE can effectively reduce the environmental disturbance to its lowest level. The experimental results of the CLPE match well with those of HP5529A for both short and long ranges. The CLPE can measure 2D displacement with high resolutions of 0.07 ± 0.021 nm and 0.07 ± 0.023 nm in the X - and Y - axes and also presents high system stabilities of −0.59 ± 0.43 nm/h and −0.63 ± 0.47 nm/h respectively in the X - and Y - axes. The CLPE has promising potential for nanometer resolution and large-range applications.

© 2013 OSA

1. Introduction

Two-dimensional (2D) or planar displacement measurements play an important role in precision equipment of a variety of scientific and technology fields, such as scanning probe microscopes (SPM), scanning electron microscopes (SEM), stepping lithographers, and so on. Subnanometer resolution, long-range measurement, and high throughput are all necessary for this coming technology node. For subnanometer resolution and long-range measurement, the noise level and drift of a measurement system must be small enough. Practically, displacement drift error in a measurement system is frequently more than hundreds of nm per hour [1

1. C. C. Wu, J. S. Yang, C. Y. Cheng, and Y. Z. Chen, “Common-path laser encoder,” Sens. Actuat. A 189, 86–92 (2013). [CrossRef]

]. Note that the error contribution of drift generally cannot be averaged out using a number of measurements identically carried out over a reasonable time [2

2. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981). [CrossRef] [PubMed]

5

5. V. V. Yashchuk, “Optimal measurement strategies for effective suppression of drift errors,” Rev. Sci. Instrum. 80(11), 115101 (2009). [CrossRef] [PubMed]

]. In addition, environmental disturbances, including temperature, pressure, and humidity variations, as well as vibrations from the base and foundation, directly contribute an additional phase change to the interference signals. The displacement error from such an additional phase change can rise to the micron range, which can be about three orders of magnitude greater than those errors from optics nonlinearity [6

6. C.-M. Wu and R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37(28), 6696–6700 (1998). [CrossRef] [PubMed]

]. However, such an additional phase change in laser encoders and laser interferometers cannot be effectively corrected through software or electronic processing. Thus, the error from environmental disturbances is the barrier to subnanometer resolution in a displacement measurement system.

Many publications over the past decade have studied planar laser encoders [7

7. W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003). [CrossRef]

20

20. C.-C. Hsu, M.-C. Kao, K.-C. Huang, and C.-C. Wu, “Reflection type displacement sensor with volume hologram for in-plane displacement measurement,” in 2012 International Conference on Measurement, Information and Control (MIC) (2012), pp. 13–16. [CrossRef]

]. Gao et al. studied a series of surface encoders for measurement of in-plane X- and Y- displacements as well as yaw and out-of-plane motions. These surface encoders capable of multi-axial measurements concentrated on the error measurements in precision stages and CNC machine tools [7

7. W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003). [CrossRef]

14

14. X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013). [CrossRef]

]. Kao et al. reported a homodyne planar laser encoder with an optical configuration of four symmetric 1-x telescopes [15

15. C. F. Kao, S. H. Lu, and M. H. Lu, “High resolution planar encoder by retro-reflection,” Rev. Sci. Instrum. 76(8), 085110 (2005). [CrossRef]

]. The measurement resolution of the planar laser encoder was 1 nm under an electronic interpolation with a factor of 400. A circular motion experiment indicated that the motion deviation of the planar encoder was less than 30 nm and the repeatability was better than 8 nm. Chung et al. proposed a planar diffraction grating interferometer (PDGI) [16

16. Y.-C. Chung, K.-C. Fan, and B.-C. Lee, “Development of a novel planar encoder for 2D displacement measurement in nanometer resolution and accuracy,” in Intelligent Control and Automation (WCICA), 2011 9th World Congress on(2011), pp. 449–453.

]. The PDGI combined two independent sets of linear diffraction grating interferometers (LDGIs) having independent optical paths and laser sources. Its measurement errors in the X- and Y- directions were better than 20 nm for a 25-mm range in a common laboratory environment. Hsieh et al. presented a 2D quasi-common-path heterodyne grating interferometer (QCOP) for X- and Y- displacement measurements in the few-millimeter range [17

17. H.-L. Hsieh, J.-C. Chen, G. Lerondel, and J.-Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011). [CrossRef] [PubMed]

]. Two types of 2D QCOPs were investigated. The corresponding measurement resolutions could be estimated to be 1.41 and 2.52 nm respectively. Lee et al. also studied 2D QCOP for 2D displacement and straightness measurements [18

18. J.-Y. Lee, H.-L. Hsieh, G. Lerondel, R. Deturche, M.-P. Lu, and J.-C. Chen, “Heterodyne grating interferometer based on a quasi-common-optical-path configuration for a two-degrees-of-freedom straightness measurement,” Appl. Opt. 50(9), 1272–1279 (2011). [CrossRef] [PubMed]

]. It achieved a system resolution and stability of 1.3 nm and 40 nm, respectively, within a 30-min measurement. In the above-mentioned publications, only Hsieh et al. and Lee et al. considered the problem of environmental disturbances.

2. Principle

A 2D grating scale is installed and fixed on the base of a moving stage. The optical read-head of the CLPE is installed on the moving platform of the moving stage. The optical configuration of the CLPE is shown in Fig. 1
Fig. 1 The optical configuration of the CLPE.
. Actually, there are two common-path configurations in the CLPE readhead [21

21. C.-C. Wu, C.-H. Liao, Y.-Z. Chen, and J.-S. Yang, “Common-path Laser Encoder with Littrow Configuration,” Sens. Actuat. A 193, 69–78 (2013). [CrossRef]

]. One configuration is for the X-displacement, and the other one is for the Y-displacement. A laser beam passes through a Fresnel lens and is focused on a two-dimensional grating scale, with the focal point on the grating scale. The grating scale diffracts the focused laser beam and builds up a three-dimensional (3D) diffraction, with the two-dimensional diffraction orders represented by (m,n), where m and n are integers. The interference between the (0, 0) and (1, 0) order diffracted beams, and that between the (0, 0) and (0, 1) order diffracted beams, must meet the condition X+λ/ρi}>0, where i=X,Y and Ω represents the convergent angle of the focused beam (see Fig. 1). With the help of this inequality, we can estimate the overlap angle of the common-path region. For example, a common numerical aperture of 0.3 has an overlap angle of 10.6°. Such an overlap angle is much larger than those in the works of Lee et al. and Hsieh et al. [17

17. H.-L. Hsieh, J.-C. Chen, G. Lerondel, and J.-Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011). [CrossRef] [PubMed]

, 18

18. J.-Y. Lee, H.-L. Hsieh, G. Lerondel, R. Deturche, M.-P. Lu, and J.-C. Chen, “Heterodyne grating interferometer based on a quasi-common-optical-path configuration for a two-degrees-of-freedom straightness measurement,” Appl. Opt. 50(9), 1272–1279 (2011). [CrossRef] [PubMed]

]. Based on the momentum conservation, the three-dimensional diffraction equations can be written as follows [22

22. R. Petit and L. C. Botten, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

]:
kXmn=k0sinθcosϕ+2mπρX,
(1)
kYmn=k0sinϕ+2nπρY,
(2)
kZmn=k02kXmn2kYmn2,
(3)
where kXmn, kYmn, and kZmn are the X-, Y- and Z- components of the wavevector kmn of the (m,n) order diffracted beam, ρX and ρY are the grating pitches along the X- and Y- directions respectively, and k0=2π/λ. The coordinate system of the three-dimensional diffraction is shown in Fig. 2
Fig. 2 Schematic of the coordinate system of the three-dimensional diffraction for a 2D grating scale. θ represents the angle of incidence for a wavevector kin with respect to the Z-axis, and ϕ is the azimuth angle of the incidence plane with respect to the X-axis. ρX and ρY are the grating pitches of a grating scale along the X- and Y- axes. O is the origin of the XYZ coordinate system. h is the grating depth.
. Note that the X- and Y-axes coincide with the grating vectors of a 2D grating scale for the 3D diffraction equations.

2.1 Doppler frequency shift

2.2 Two-aperture phase shifting technique

3. Runout tolerance analysis

A planar encoder has only three types of runout: yaw, tilt, and standoff. These three types of runout come from the runout tolerance of the moving stage. The tolerances for a common stage are typically near 5 arc-sec (2.4×105 rad) and 0.4 μm for yaw/tilt and standoff tolerances [24]. The works of Kao et al. [15

15. C. F. Kao, S. H. Lu, and M. H. Lu, “High resolution planar encoder by retro-reflection,” Rev. Sci. Instrum. 76(8), 085110 (2005). [CrossRef]

, 25

25. C.-F. Kao, C. C. Chang, and M.-H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng. 44(2), 023603 (2005). [CrossRef]

] adopted the configuration with the 2D grating scale as a movable component. Physically, the runout of the grating scale may make the coordinate system of the 2D grating scale not coincide with the coordinate system of the grating velocity. Thus, a coordinate transformation must be considered in their analysis, and the displacement measurements along the X- and Y- axes do couple to each other for their planar encoder. For the CLPE, we adopt the configuration with the optical read-head of the CLPE as a moveable component. As the coordinate systems of the moveable optical read-head and the 2D grating scale always coincide, the CLPE can attain a decoupling 2D displacement measurement between the displacements along the X- and Y- axes. In the following, we will apply the first-order analysis to estimate runout tolerance of the CLPE.

First, consider a case of yaw runout at ψ angle for the CLPE (see Fig. 1). The interference fringe will have a displacement with respect to the apertures, A1, A2, A3, and A4. We can estimate the yaw runout tolerance based on the following inequality:
D2ψq10000,
(12)
where D is the diameter of the laser beam (see Fig. 1). Note that the factor of 10,000 can ensure the displacement error from runout smaller than (a grating pitch)/10,000. As q and D are about 2.5 mm and 10 mm respectively, we can have the estimated yaw runout tolerance near 5×105 rad (10.3 arc-sec). Such a yaw runout tolerance is enough and ensures a displacement measurement error within 0.16 nm (if the grating pitch ρX=ρY=1.6 μm).

Second, consider a case of tilt runout for the CLPE. As the effects of the X-tilt and the Y-tilt of the CLPE are the same, we can consider X-tilt only; that is, θ = 0° and φ = τ. The interference fringe will have a displacement with respect to the apertures A1, A2, A3, and A4. We can estimate the tilt runout tolerance based on the following inequality:
12cos1(k10k10//k10k10k10//k10//)(D2)2+f2q10000,
(13)
where k10// represents the projection vector of k10 in the X-Z plane, and f is the focal length of the Fresnel lens. Similarly, we adopt the same factor of 10,000 as Eq. (12). As q, D and f are about 2.5 mm, 10 mm, and 16.67 mm respectively, we can obtain the estimated tilt runout tolerance τ2.87×105 rad (5.93 arc-sec). Such a tilt runout tolerance is enough and ensures a displacement measurement error smaller than 0.16 nm (if the grating pitch ρX=ρY=1.6 μm).

Third, consider a case of standoff runout at S μm for the CLPE. The depth of focus for a common lens with the f-number of 1.67 is about 3.5 μm [26

26. F. L. Pedrotti, L. M. Pedrotti, and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, 2006).

]. Such a depth of focus is much larger than the standoff runout of the stage. Thus, the standoff tolerance for the CLPE is S = 3.5 μm, and standoff runout within such a tolerance cannot induce additional measurement error for the CLPE.

4. Experiments and results

Figure 4
Fig. 4 Schematic of experimental setup with the CLPE configuration built in the readhead.
shows the schematic of the experimental setup in this study with the CLPE configuration built in the readhead. A frequency-stabilized He-Ne laser source (model HRS015, Thorlabs, Inc.) with a wavelength of 632.991 nm (in vacuum) was used to provide a good light source. Since the CLPE is common-path in its configuration, a low coherence light source, such as a diode laser, can be used for compactness. A 2D grating scale (manufactured by Taiwan Mask Corp.) with grating pitches of 1.6 μm had two grating vectors coinciding respectively with the X- and Y- axes. The dimensions of the grating scale were 100 mm × 100 mm. The grating scale was mounted on an optical table (model: RS4000- 412 −12, Newport Inc.). The substrate of the grating was borosilicate glass with a thermal expansion coefficient of 3.3×106/C. To provide long- and small-range 2D displacements, a 2D composite stage was made of two types of 2D linear stages: two long-ranged motorized linear stages (model: SGSP26-200, Sigma Inc.) with a controller (model: SHOT-702, Sigma Inc.) and a small-ranged piezoelectric stage (model: Tritor 100 SG, Piezosystem Jena) with a controller (model: NV40/3 CLE, Piezosystem Jena). The small-range piezoelectric stage used three embedded strain gauges as its displacement sensors [27]. The CPLE was used to measure the displacement of the 2D composite-stage.

4.1 Long-range measurement

The long-range measurement results of the HP5529A and the CLPE in a square path with each side at 8 mm are shown in Fig. 5
Fig. 5 Results of 8-mm square path movement by the HP5529A and the CLPE.
. The results of the HP5529A and the CLPE match well. At corners of the square path, there exist about 50-nm discrepancies between the result of the HP5529A and the CLPE. As such discrepancies fall to near 10 nm at the starting position of the stage, we infer that such discrepancies come mainly from the cosine error between the measurement axes of the HP5529A and the CLPE. Certainly, the drift effect contributes to these discrepancies. The drift issue will be discussed in section 4.3, where we address stability.

4.2 Short-range measurement

(i) 20-μm triangular movement

Figure 6
Fig. 6 Results of 20-μm triangular path movement by the HP5529A, the CLPE, and the strain gauge.
demonstrates the results of 20-μm triangular path movement by the HP5529A, the CLPE, and the strain gauge built in the piezoelectric stage. These three results match well. Three insets of Fig. 6 show details of the discrepancies between the three methods. From these insets, we can find a larger difference in the result of the HP5529A from those of the CLPE and the strain gauge. Such a difference comes mainly from the resolution truncation of the HP5529A (10-nm resolution).

(ii) 10-nm diameter circular movement

In order to investigate the short-range measurement capability of the CLPE, we drove the piezoelectric stage using two analog voltage inputs to move the stage in a circular path with a diameter of 10 nm under the open-loop mode. Figure 7
Fig. 7 Results of 10-nm diameter circular movement by the strain gauge and the CLPE.
represents the results of 10-nm diameter circular movement by the CLPE and the strain gauge built into the piezoelectric stage. These two results have good matching in the trend. However, the data scatter of the strain gauge has a much larger range than that of the CLPE. Such a data scatter comes from the resolution limit of the strain gauge (about 3 nm). In contrast, the CLPE has a subnanometric resolution and a promising potential for subnanometric applications. Due to high-frequency noise, the dashed line of the CLPE looks like a “black bold” line. The resolution of the CLPE will be discussed in section 5.2.

4.3 Stability measurement

Stability is an important specification to a common-path laser encoder or grating interferometer. To verify the system stability of the CLPE, which reflects its immunity to environmental disturbances, we immobilized both the coarse and fine stages carrying the grating scale for three hours. The two drivers of the composite stage were turned off electrically. We used the CLPE and the HP5529A to simultaneously monitor the X- and Y- displacement variations of the grating scale. Figure 8
Fig. 8 System stability measurement results of the CLPE and the HP5529A for three hours.
shows the system stability measurement results of the CLPE and the HP5529A for three hours. We’ve deliberately shifted the HP5529A data by −500 nm for convenient observation. Results show that the CLPE data had stabilities of −0.59 ± 0.43 nm/h and −0.63 ± 0.47 nm/h respectively in the X- and Y- axes, which were over two orders of magnitude smaller than those indicated by the HP5529A data (−118.67 ± 49.50 nm/h and −120 ± 50.00 nm/h respectively). The CLPE data physically reflected the state of the still stage. The insets of Fig. 8 show the X- and Y- data of the CLPE from 5400 s to 5508 s. In addition, the system drift data of the CLPE in the X- and Y- direction were two orders of magnitude smaller than those of the HP5529A within three hours.

5. Discussion

5.1 Quadrature error

5.2 Resolution

In Eqs. (10) and (11), two measured phases suffer from various high- and low-frequency noises. Though the CLPE has a theoretical displacement measurement resolution of about 24 pm for grating pitches of 1.6 μm, these noises spoil the measurement resolutions. Because the stage is held still (d(ΔΦX) = d(ΔΦY) = 0), the effect of the grating pitch variations or pitch tolerances on the measurement error can be dropped. That is, the results in Fig. 8 indicate only the phase noises. We can estimate the measurement resolution of the CLPE using the stability data [21

21. C.-C. Wu, C.-H. Liao, Y.-Z. Chen, and J.-S. Yang, “Common-path Laser Encoder with Littrow Configuration,” Sens. Actuat. A 193, 69–78 (2013). [CrossRef]

]. The measurement times for all short- and long-range experiments in this study were within 600 s. A review of every 600-s time interval in Fig. 8 shows two estimated measurement resolutions of 0.23 ± 0.074 nm and 0.25 ± 0.080 nm, respectively, in the X- and Y- axes. For short-range measurement, the review time interval can be as low as 60 s, and two greatly superior estimated measurement resolutions of 0.07 ± 0.021 nm and 0.07 ± 0.023 nm in the X- and Y- axes can be obtained. In contrast, the drift of the HP5529A is large, so the corresponding estimated measurement resolutions respectively in the X- and Y- axes within 600 s are 130.0 ± 18.6 nm and 130.7 ± 18.6 nm, and within 60 s are 80.85 ± 16.61 nm and 82.01 ± 16.36 nm, respectively. The major difference of estimated resolutions between the CLPE and the HP5529A comes from the drift effect. Although the HP5529A has a resolution of 10 nm listed on its data sheet, practically, we cannot attain such a resolution in a common engineering environment, since the drift effect engulfs the resolving capability of the HP5529A. Note that the resolution of a laser encoder or an interferometer is not a constant value and depends on the measurement time interval, since drift exists.

6. Conclusion

We propose an innovative CLPE for 2D displacement measurement. As the CLPE is common-path, it can effectively overcome the environmental disturbance. The CLPE adopts a 1-x telescope configuration, so its runout tolerances are large enough to cope with the problem of practical runout. An analysis method for runout tolerances of the CLPE is presented. We verify the performance of the CLPE for short- and long-range measurements. The experimental results demonstrate that the CLPE not only can measure 2D displacement with high resolutions of 0.07 ± 0.021 nm and 0.07 ± 0.023 nm in the X- and Y- axes (for a short time interval of 60 s), but also presents high system stabilities of −0.59 ± 0.43 nm/h and −0.63 ± 0.47 nm/h respectively in the X- and Y- axes.

Acknowledgments

This study was supported by the National Science Council of Taiwan under contracts NSC 100-2628-E-032- 001- and NSC 102-2221-E-032-012-MY3. The authors cordially thank Prof. C.-K. Lee of National Taiwan University for his assistance.

References and links

1.

C. C. Wu, J. S. Yang, C. Y. Cheng, and Y. Z. Chen, “Common-path laser encoder,” Sens. Actuat. A 189, 86–92 (2013). [CrossRef]

2.

P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981). [CrossRef] [PubMed]

3.

J. D. Ellis, M. Baas, K.-N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng. 36(2), 261–269 (2012). [CrossRef]

4.

W.-W. Chiang and C.-K. Lee, “Wavefront reconstruction optics for use in a disk drive position measurement system,” USPTO, ed. (International Bussiness Machines, America, 1995).

5.

V. V. Yashchuk, “Optimal measurement strategies for effective suppression of drift errors,” Rev. Sci. Instrum. 80(11), 115101 (2009). [CrossRef] [PubMed]

6.

C.-M. Wu and R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37(28), 6696–6700 (1998). [CrossRef] [PubMed]

7.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003). [CrossRef]

8.

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor-driven planar motion stage integrated with an XYθZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004). [CrossRef]

9.

W. Gao, S. Dejima, and S. Kiyono, “A dual-mode surface encoder for position measurement,” Sens. Actuat. A 117(1), 95–102 (2005). [CrossRef]

10.

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Annals - Manufacturing Technology 56(1), 529–532 (2007). [CrossRef]

11.

A. Kimura, W. Gao, A. Yoshikazu, and L. Zeng, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010). [CrossRef]

12.

A. Kimura, W. Gao, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol. 21(5), 054005 (2010). [CrossRef]

13.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012). [CrossRef]

14.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013). [CrossRef]

15.

C. F. Kao, S. H. Lu, and M. H. Lu, “High resolution planar encoder by retro-reflection,” Rev. Sci. Instrum. 76(8), 085110 (2005). [CrossRef]

16.

Y.-C. Chung, K.-C. Fan, and B.-C. Lee, “Development of a novel planar encoder for 2D displacement measurement in nanometer resolution and accuracy,” in Intelligent Control and Automation (WCICA), 2011 9th World Congress on(2011), pp. 449–453.

17.

H.-L. Hsieh, J.-C. Chen, G. Lerondel, and J.-Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011). [CrossRef] [PubMed]

18.

J.-Y. Lee, H.-L. Hsieh, G. Lerondel, R. Deturche, M.-P. Lu, and J.-C. Chen, “Heterodyne grating interferometer based on a quasi-common-optical-path configuration for a two-degrees-of-freedom straightness measurement,” Appl. Opt. 50(9), 1272–1279 (2011). [CrossRef] [PubMed]

19.

K. C. Fan, B. H. Liao, Y. C. Chung, and T. T. Chung, “Displacement measurement of planar stage by diffraction planar encoder in nanometer resolution,” in 2012 IEEE International Conference on Instrumentation and Measurement Technology(2012), pp. 894–897. [CrossRef]

20.

C.-C. Hsu, M.-C. Kao, K.-C. Huang, and C.-C. Wu, “Reflection type displacement sensor with volume hologram for in-plane displacement measurement,” in 2012 International Conference on Measurement, Information and Control (MIC) (2012), pp. 13–16. [CrossRef]

21.

C.-C. Wu, C.-H. Liao, Y.-Z. Chen, and J.-S. Yang, “Common-path Laser Encoder with Littrow Configuration,” Sens. Actuat. A 193, 69–78 (2013). [CrossRef]

22.

R. Petit and L. C. Botten, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

23.

L. E. Drain, The Laser Doppler Technique (John Wiley, 1980).

24.

http://www.newport.com.

25.

C.-F. Kao, C. C. Chang, and M.-H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng. 44(2), 023603 (2005). [CrossRef]

26.

F. L. Pedrotti, L. M. Pedrotti, and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, 2006).

27.

http://www.piezosystem.com/home/.

28.

P. Gregorcic, T. Pozar, and J. Mozina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17(18), 16322–16331 (2009). [CrossRef] [PubMed]

29.

Sony Precision Technology, http://www.sonypt.com/.

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.3180) Instrumentation, measurement, and metrology : Interferometry

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: June 5, 2013
Revised Manuscript: July 4, 2013
Manuscript Accepted: July 5, 2013
Published: August 1, 2013

Citation
Chyan-Chyi Wu, Yan-Zou Chen, and Chia-Huang Liao, "Common-path laser planar encoder," Opt. Express 21, 18872-18883 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-18872


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References

  1. C. C. Wu, J. S. Yang, C. Y. Cheng, and Y. Z. Chen, “Common-path laser encoder,” Sens. Actuat. A189, 86–92 (2013). [CrossRef]
  2. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt.20(19), 3382–3384 (1981). [CrossRef] [PubMed]
  3. J. D. Ellis, M. Baas, K.-N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng.36(2), 261–269 (2012). [CrossRef]
  4. W.-W. Chiang and C.-K. Lee, “Wavefront reconstruction optics for use in a disk drive position measurement system,” USPTO, ed. (International Bussiness Machines, America, 1995).
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