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Common-path laser encoder
Sensors and Actuators A 189 (2013) 86–92
Contents lists available at SciVerse ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Common-path laser encoder Chyan-Chyi Wu ∗ , Jhih-Sheng Yang, Chun-Yao Cheng, Yan-Zou Chen Advanced Photonics Laboratory, Department of Mechanical and Electro-mechanical Engineering, Tamkang University, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 21 June 2012 Received in revised form 24 August 2012 Accepted 25 August 2012 Available online 5 September 2012 Keywords: Common-path Grating Displacement Long-range Nanometer
a b s t r a c t This study proposes a novel common-path laser encoder (CPLE) capable of effectively minimizing environmental disturbance. The proposed CPLE uses a two-aperture phase-shifting technique to form quadrature signals. Experimental results match well with HP5529A results for long-range measurements. Results also show that the estimated measurement resolution is 0.1 ± 0.046 nm. Therefore, the proposed design has great potential for nanometer resolution and long-range applications. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Displacement measurements play an important role in the precise positioning systems and equipment of semiconductor and liquid crystal display (LCD) manufacturing. With the rapid advance of nanotechnology, 32 nm node on a 450 mm wafer should be feasible in 2012 [1]. Subnanometer resolution and superior immunity to environmental disturbance are required to facilitate displacement measurements over such a long range. Researchers have developed many displacement measurement techniques over the past few years. Optical interferometry is the most important of the various displacement measurement techniques. Optical interferometers are widely used for the precise measurement of displacement and other physical parameters because they have the capability of a long range and high resolution [2]. However, variations in temperature, pressure, humidity and vibration caused by environmental disturbance reduce measurement accuracy. Consequently, environmental conditions must be strictly controlled to achieve high measurement accuracy. The wavelength of the laser source also affects measurement accuracy [3–5]. Unlike optical interferometers, grating interferometers or laser encoders are not subject to environmental disturbances. These devices have better immunity to the environmental disturbances than optical interferometers because they transfer the measurement scale from the laser wavelength into the grating pitch. Teimel [6] introduced several laser encoders for submicron applications,
∗ Corresponding author at: Department of Mechanical and Electromechanical Engineering, Tamkang University, Tamsui Dist., New Taipei City 25137, Taiwan, ROC. E-mail addresses: [email protected], [email protected] (C.-C. Wu). 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.08.034
reaching a measurement resolution of 5 nm. Dobosz [7] demonstrated a laser encoder with a resolution of 10 nm. He also proposed a method to slash the optics nonlinearity of laser encoders. Nevière et al. [8] used two gratings to detect linear displacement, and the laser encoder had a measurement resolution of 80 nm. Lee et al. [9] developed an optical heterodyne laser encoder with a transmissive grating. This laser encoder is robust to environmental disturbances. Wu et al. [10] adopted a laser encoder with a perpendicular optical configuration. This laser encoder has high head-to-scale tolerance and an accuracy of 37.3 nm. Kao et al. [11] presented a laser encoder with Littrow configuration grating. This encoder achieves a maximum measurement error of 53 nm and repeatability within ±20 nm for a 100 m measurement range. Although these laser encoders achieve nanometric resolution, their optical configurations remain in a non-common path. Lee et al. [12,13] presented a quasi-common optical path laser encoder with a heterodyne laser source. They adopted the configuration of polarization interferometry, and used a half-waveplate to extract the p-polarization light from the spolarization light before modulating the measured quantity into the laser source. In this design, the p- and s-polarization light traverse side by side, but not along the same path. This device achieves a positioning resolution of 2.3 nm over a 20 mm range. However, no researcher has yet developed a common-path laser encoder. Lee’s work would be state of the art for a laser encoder because its configuration is closest to the common-path interferometer. The main problem with a common-path configuration for Lee’s laser encoder is how to perform phase shifting in the same optical path, especially after the reference and test beams interfere. This study presents a CPLE with a simple optical configuration. Because the CPLE is common-path, it possesses high measurement resolution and immunity to environmental disturbances. The CPLE
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Fig. 1. Optical configuration of the CPLE and experimental setup.
adopts a two-aperture phase-shifting technique to achieve phase shifting after the reference beam interferes with the test beam in the same optical path. This study also presents a detailed description of the CPLE operating principle. Experimental results show that the CPLE has a resolution of 0.1 ± 0.046 nm. These results demonstrate that the CPLE can measure short and long displacement with nanometric resolution. 2. Principle Fig. 1 shows the optical configuration of the CPLE. A laser source (LS) is focused by lens L1 onto the diffraction grating G and then split into 0th order and −1st order diffraction beams. These two diffracted beams diverge and overlap partly in the adjacent regions. These two divergent beams are collimated by lens L2 and then pass through the aperture stop As to extract the overlapping region of the 0th and −1st order diffracted beams (see the inset of Fig. 1). After As , these two collimated beams produce typical interference fringe. These two beams subsequently enter the beam splitter BS and pass through apertures A1 and A2 , respectively. The apertures A1 and A2 can assign an additional phase shifting for the interference signals in the CPLE or interferometers with a common path. Finally, photodetectors PD1 and PD2 measure two-beam interference fringes. This section introduces the Doppler frequency shift, which is modulated into the diffracted beams when the diffraction grating G is moving. This section also presents the principle of phase shifting using two apertures and derives the relationship between the phase variation and the grating displacement. 2.1. Doppler frequency shift in focus/divergent beam The displacement of diffraction grating G modulates a specific Doppler frequency shift ωm into each diffracted beam. This relationship can be written as [10]. ωm = vG · (k m − k i ),
(1)
where m is the diffraction order, vG is the velocity of diffraction grating G, k m is the wavevector of the mth diffraction order beam, and k i is the wavevector of the incident beam. In (1), the Doppler frequency shift depends only on the difference of the horizontal projections of the wavevectors between the incident and diffracted beams for a given velocity vG of diffraction grating G. For convenience, consider the in-plane diffraction cases shown in Fig. 2. Because the incident beam focuses onto diffraction grating G, there are separate wavevectors for various parts of the incident beam. Similarly, there are also separate wavevectors for the various parts of the specific diffraction order (divergent) beam. Consider that the focus and divergent beams comprise many finite beams, as Fig. 2 shows. The grating equation must hold for every finite beam pair in
Fig. 2. Finite beam model and schematics of different horizontal projections of wave vectors.
the focus or divergent beams. The grating equation, in the momentum conservation form, can be written as follows:
2m k m − k = , i
(2)
where is the grating pitch. Eq. (2) shows that the difference between the horizontal projections of the incident wavevector and diffracted wavevector is equal to the corresponding grating momentum, 2m/. Eq. (2) also holds for the linear grating diffraction in three dimensions. Combining (1) and (2), the Doppler frequency shift can be rewritten as follows: ωm = sign(vG )
2m |vG |,
(3)
where the sign function, sign(vG ), is equal to +1 when vG is along +X, and sign(vG ) is equal to −1 when vG is along −X. Eq. (3) shows that every finite beam pair has the same Doppler frequency shift for a specific diffraction order, irrespective of whether the beam is focused or divergent. 2.2. Interference condition for diffracted beams The proposed CPLE uses the divergent 0th and −1st order diffracted beams to interfere with each other. However, a certain condition must be met to successfully obtain the interference fringe. Consider the schematic of the overlap region of the 0th and −1st diffracted beams (Fig. 3). The grating equation reads [14] sin d = sin i +
m ,
(4)
where d is the diffraction angle of the mth order diffracted beam, i is the incident angle of the laser beam, m is the diffraction order, is the grating pitch, and is the wavelength of the laser source. Eq. (4) identifies the key diffracted angle of each marginal ray of the focusing laser beam; that is, 0 and u−1 . Fig. 3 shows that u−1 must be greater than 0 to achieve a proper overlap region between the 0th
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Fig. 3. The schematic of the overlap region of the 0th and 1st diffracted beams, where represents the convergent angle of the focusing laser beam, l represents the lower incident angle of the marginal rays, and u represents the upper incident angle of the marginal rays.
Fig. 5. A1 and A2 are aligned with the direction of the interference fringe. The term a is the width of A1 and A2 , and is the relative distance between A1 and A2 in interference fringe space.
and −1st order diffracted beams. That is, the following inequality must be met:
√ where A0 is the amplitude of E0 , and j is equal to −1. Likewise, the , can be written electric field of the −1st-order diffracted beam, E−1 as follows:
sin−1 sin( + ) −
− > 0.
(5)
Here 0 = 1 is substituted into (5), which can then be used to estimate the overlap angle of the 0th- and −1st-order diffracted beams. Fig. 4 shows the simulation results for three convergent angles of the focusing laser beam: 25◦ , 30◦ , and 50◦ . Fig. 4 shows that a large convergent angle (i.e., a lens with large numerical aperture) is helpful for the overlap of the diffracted beams. Conversely, a large angle of incidence (similar with ) is not suitable for overlap of the diffracted beams. The minimum convergent angle to achieve an overlap region of the diffracted beams is approximately 25◦ . If the 0th- and +1st-order diffracted beams interfere, the overlap angle is slightly larger than choosing the 0th- and −1st-order diffracted beams. Because the difference between these two cases is small, the proposed CPLE uses the 0th- and −1st-order diffracted beams for experimental convenience.
E−1 = A−1 ejω−1 t
1 0
,
(7)
where ω−1 represents the Doppler frequency shift of the −1st-order . These two fields, diffracted beam, and A−1 is the amplitude of E−1 , interfere with each other, and their interference fringe E0 and E−1 can be written as follows: I = A20 + A2−1 + 2A0 A−1 cos
2 q
x − ω−1 t ,
(8)
where q is the spatial period of the fringe, and x is the local coordinate for the interference fringe. Both A1 and A2 are aligned with the direction of the interference fringe (Fig. 5). By integrating (8) over the ranges (0, a) and ( , a + ), respectively, we can represent the output signals I1 and I2 of the detectors PD1 and PD2 as follows: I1 ∝ cos
a q
− ω−1 t ,
(a + 2 )
(9)
2.3. Phase shifting using two apertures
I2 ∝ cos
For the proposed CPLE (Fig. 1), the electric field of the 0th-order diffracted beam after passing L2, E0 , can be represented as follows:
where a is the width of A1 and A2 , and is the relative distance between A1 and A2 in the interference fringe space. If 2/q = (2n − 1)/2 with n belonging to the natural number in (10), we can obtain I1 and I2 in quadrature. The condition 2/q = (2n − 1)/2 can be achieved easily by changing according to the Lissajous trace of I1 and I2 signals. The common phase term in (9) and (10), a/q, can be removed in the process of phase unwrapping and the displacement calculation because the CPLE is an incremental type laser encoder. Thus, the CPLE realizes the phase shifting after the reference beam interferes with the test beam for common-path interferometers. Combining (3), (9) and (10) shows the relationship between the grating displacement X and the measured phase change ˚ as the following expression:
E0 = A0
1 0
,
(6)
X =
q
− ω−1 t ,
˚. 2
(10)
(11)
Eq. (11) shows that the grating displacement depends on the grating pitch and phase change. This equation also shows that the grating displaces a pitch distance for a full cycle of the quadrature signal. 3. Experiments and results
Fig. 4. The simulation results for three convergent angles of the focusing laser beam: 25◦ , 30◦ , and 50◦ .
Fig. 1 shows the experimental setup in this study. A frequencystabilized He–Ne laser source (model HRS015, Thorlabs, Inc.) with a wavelength of 632.991 nm (in vacuum) was used. A grating
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Fig. 6. The measured results for the stability.
Fig. 7. Measurement results of 10.8 mm displacement at different stage velocities.
with a grating pitch of 1.6 m (model: THOR-GT50-06V, Thorlabs, Inc.) was mounted on a homemade composite stage (CS), and this grating diffracted the laser source. The substrate of the grating is borosilicate glass with a thermal expansion coefficient of 3.3 × 10−6 /◦ C. To provide long- and small-range displacements, the composite stage CS was made of two types of linear stages: a longranged motorized linear stage (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 optical layout was placed on an isolated optical table (model: RS4000-412-12, Newport Inc.) to reduce the low-frequency mechanical noise from the foundation. The experiments were divided into three parts: (i) stability measurement; (ii) long-range measurement; and (iii) small-range measurement. The CPLE was used to measure the displacement of the grating or the composite-stage CS. The CPLE results were then compared with those of a commercial laser interferometer (HP5529A). The laboratory atmosphere conditions were monitored using the HP5529A sensors. All experiments were implemented under the same temperature control condition. The average atmosphere temperature was 25.4 ◦ C, with a standard deviation of 0.32 ◦ C. The average atmospheric pressure was 761.6 mmHg, with a standard deviation of 0.6 mmHg. With the thermal expansion coefficient 3.3 × 10−6 /◦ C given, for the worst case the temperature changes 2 ◦ C, the grating pitch (1.6 m) will elongate about 1.1 pm. And the effect of the grating pitch thermal expansion may linearly increase with the increase of the measured range. As our experimental results are for the travel range smaller than 10.8 mm, the error due to thermal expansion of the grating pitch will be smaller than 8 nm.
11 nm and 250 nm variations, respectively. Fig. 6 shows that the CPLE is much less sensitive to the surrounding disturbance than the HP5529A. This figure also explains why an optical interferometer is an improper candidate for nanoscale displacement measurements in engineering environments. The CPLE provides superior system stability and can efficiently reduce the influence of environmental noises.
3.1. Stability measurement Immunity to environmental disturbances can dramatically influence the applications of the proposed laser encoder. To verify the stability of the CPLE, which reflects its immunity to environmental disturbance, the grating and composite stage were held stationary in laboratory coordinates for 2 h (7200 s). We used HP5529A to monitor the stationary grating simultaneously. The experimental setup was the same as that in Fig. 1. Fig. 6 shows the measured results for stability. This figure enables estimating the long-term stability of the CPLE and HP5529A. The CPLE data truly reflect the state of the still stage. Over 2 h, there were approximately
3.2. Long-range measurement First, the target stage displacement was set at 10.8 mm based on the reading of the HP5529A, and the CPLE was used to measure the stage displacement. The sampling rate of the HP5529A was 10 Hz. Fig. 7 shows the measurement results of 10.8 mm displacement at different stage velocities. The dashed line in this figure was deliberately biased by 200 s on the time axis to differentiate between the CPLE and HP5529A data. To investigate the measurement repeatability of the CPLE for long-range measurements, we repeated the experiment of 10.8 mm displacement five times, with a stage velocity of 6.1 m/s. Table 1 shows the displacements measured by the CPLE and HP5529A for the 10.8 mm range. The measured results of the HP5529A and CPLE exhibit a few differences. The overall discrepancy was approximately 614 nm for five measurements, and the relative deviation was approximately 0.057/1000. In these experiments, the cosine error was estimated at 0.038/1000, that is, its contribution was approximately 411 nm if the included angle for the cosine error is 0.5◦ , which is frequently encountered. According to Fig. 6, with a measurement time of 1800 s, the measurement deviation between CPLE and HP5529A was approximately 100 nm. The left deviation may partly come from the cyclic nonlinearity error. According to Wu’s analysis, the typical cyclic error is about a few nanometers [15]. The manufacture tolerance of the grating may contributes partly to the measurement deviation. Generally, the grating tolerance allocates 10% measurement accuracy for a Table 1 The displacements measured by the CPLE and HP5529A for the 10.8 mm range. No.
CPLE (m)
HP5529A (m)
Deviation (nm)
Deviation ratio (%)
1 2 3 4 5
10799.615 10799.766 10799.665 10799.767 10746.912
10800.120 10800.600 10800.567 10800.364 10800.146
505 834 902 597 234
0.0047 0.0077 0.0084 0.0055 0.0022
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Fig. 8. The CPLE measurement results for the periodic triangular and sinusoidal input waveforms: (a) 5-nm range displacement and (b) 0.5-nm range displacement.
laser encoder. If an overall accuracy of 0.1 m is considered, the deviation from the grating tolerance will be 10 nm. From above analyses, we might reasonably infer that the major factor for such a discrepancy (614 nm) is caused by the cosine error (411 nm conservatively). Fortunately, better alignment can remove the cosine error. These results demonstrate that the CPLE can measure longrange displacements.
0.5 Hz and displacement amplitudes of 5 nm and 0.5 nm, respectively. Fig. 8 shows the CPLE measurement results for the periodic triangular and sinusoidal input waveforms: (i) a 5-nm range displacement and (ii) a 0.5-nm range displacement. These small-range test results indicate that the CPLE has the capability of measuring subnanometer displacements.
3.3. Small-range measurement 4. Discussion The small-range experiments in this study verified the CPLE measurement capability on the nanometer scale, especially for a displacement range of less than 10 nm. The tested ranges included 5 nm and 0.5 nm. These precise displacements were produced by moving the small-range piezoelectric stage (model: Tritor 100SG, Piezosystem Jena) of the composite stage while holding the longrange stage still. The experimental setup was as that in Fig. 1 but did not involve using HP5529A for two reasons: (i) its resolution is worse than 10 nm and (ii) its drift is too large (Fig. 6). The piezoelectric stage was driven using periodic input voltages in triangular and sinusoidal waveforms with a frequency of
Laser encoders generally suffer from two types of errors: optical path-dependent errors and component-dependent errors. Optical path-dependent errors are from misalignment (cosine and Abbe errors), vibration, temperature variation, and air turbulence. The component-dependent errors are from grating pitch tolerance, noise, and optics nonlinearity caused by polarization crosstalk. Because the CPLE is fully common-path, it essentially removes optical path-dependent errors. Because the CPLE has no polarization component, it has no optics nonlinearity error from polarization mixing. Because of its common-path configuration, the laser source
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Fig. 10. The corresponding grating pitch error is dsi for an arbitrary laser spot position Xi .
effect in operation behaves similarly to the random error. Thus, the grating pitch tolerance and the thermal effect can be combined in a random variable for subsequent analysis. Consider the total differentiation of (11) as follows:
Fig. 9. The effects of the noise level and photodetector gains on the phase change in the displacement measurement.
of the CPLE can be replaced using a short coherent length laser source, such as a compact laser diode. 4.1. Quadrature error Laser interferometers and laser encoders frequently adopt quadrature signals. The major quadrature measurement error sources are incorrect phase shift, zero offset, and gain difference [16,17]. When the phase shift between the detected signals deviates from the ideal 90◦ , the phase-unwrapping procedure generates an error in the displacement measurement. For the CPLE, the deviated interference output signals, I1 and I2 , can be expressed as I1
= (DC1 + DC1 ) + (AC1 + AC1 ) cos(˚)
(12)
and I2 = (DC2 + DC2 ) + (AC2 + AC2 ) cos(˚ + a ),
(13)
where (DCi + DCi ) and (ACi + ACi ) respectively denote the direct current and alternating current terms of the deviated intensity Ii with i = 1, 2, and a being the phase shift error between the apertures A1 and A2 . ACi causes high-frequency noises, which are mainly from the electrics. The phase shift error a in (12) and (13) can be minimized through precise alignment, and a fourphotodetector configuration can effectively remove the DC terms. Fig. 9 shows that a higher noise level spoils the measurement resolution of the CPLE. The photodetector gains also contribute periodic error to the displacement measurement. The periodic error from the photodetector gains can be reduced by fine tuning the gain of the amplification circuits. In summary, all factors of the quadrature error can be minimized through simple alignment or fine tuning the CPLE. Thus, the common periodic errors in heterodyne or homodyne laser interferometers vanish in the CPLE. 4.2. Grating pitch tolerance for different ranges The grating pitch is the measurement scale of the laser encoder. A uniform grating pitch is assumed when calculating the phase shift. However, manufacturing tolerances, incorrect installation, and the thermal effect in operation can damage the grating pitch. Thus, the grating pitch tolerance contributes to the displacement error. Manufacturing tolerances have two error sources: systematic error and random error. Systematic errors can be effectively removed using the calibration technique. Physically, the thermal
d(X) =
˚ d + d(˚). 2 2
(14)
The first term on the right side of (14) is the displacement error from the grating pitch tolerance, and the second term is the displacement error from the phase change error. The phase change error is from many factors, including optomechanical misalignment, high-frequency electric noise, optical nonlinearity, environmental disturbances, and unwrapping algorithm. Many articles have discussed these factors [9,10,17,18]. Therefore, this study concentrates on the contribution of the grating pitch tolerance. Assume an ideal case in which the phase change is error-free, that is, d(˚) = 0. Eq. (14) can be reduced as d(X) =
˚ d 2
if d(˚) = 0.
(15)
Eq. (15) shows that the displacement error is linearly proportional to the grating pitch error. However, when investigating a longrange measurement, such as 1 m, and regarding d as a constant, (15) produces an unreasonable outcome. For example, if the grating pitch = 1.6 m and the grating scale length is 1 m, then the laser spot size on the grating scale is 2 mm in diameter. A laser spot on the grating scale covers N grating lines, and N is approximately 1250. Assume that the manufacturing tolerance of the grating pitch is di for an arbitrary laser spot position Xi . Assume that the grating scale moves pitch by pitch, or cycle by cycle in interference signals. The corresponding displacement error over a pitch displacement from Xi is dsi , as shown in Fig. 10. Based on the average effect of N grating lines for a laser spot, dsi can be expressed as d dsi = √ i . N
(16)
Assume that the full measurement range is L. Within the range L, there are r times of a single pitch movement. In other words, r = [L/] + 1, where [ ] expresses the Gauss sign. The total displacement error ds can be estimated using the expression
r di 2 . ds = √ i=1
N
(17)
According to the equal accuracy principle, each pitch movement contributes an equal displacement error (i.e., ds1 = ds2 = ...). Thus, we can estimate the grating pitch error or grating pitch tolerance for different accuracies and ranges. Table 2 shows the grating pitch tolerance estimates for different measurement ranges: 30 cm, 45 cm, and 100 cm. This study uses a common commercial laser interferometer for comparison because its accuracy is a 0.1 m
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Table 2 The grating pitch tolerance estimates for different measurement ranges: 30 cm, 45 cm, and 100 cm. Measurement range (mm) 1000 450 300 200
Required accuracy (m)
Pitch tolerance (nm)
0.1 0.1 0.1 0.1
0.5 0.6 0.9 4
99-2628-E-032-001- and NSC 100-2628-E-032-001-). We also thank Professor C.K. Lee (National Taiwan University) for providing us with access to the necessary equipments. References
Because the proposed CPLE uses a 16-bit analog-to-digital converter (ADC) for displacement measurement, the theoretical displacement measurement resolution is approximately 24 pm at a grating pitch of 1.6 m. Consider the experiment shown in Fig. 1 and, when holding the stage still (i.e., d˚ = 0), the effect of the grating pitch variation can be dropped from (14). Fig. 6 shows only the phase noises. The measurement time for small-range experiments in this study is smaller than 10 s. Thus we determine the maximum and minimum data for every 10-s time interval in Fig. 6. The measurement resolution can be estimated using the difference between the maximum and minimum data. Through such a review of every 10-s time interval in Fig. 6, we can statistically obtain an estimated measurement resolution of 0.1 ± 0.046 nm. If the effective bit number for ADC is considered, the equivalent effective bit number of approximately 12 bits can be obtained. In contrast with the HP5529A, its resolution is estimated about 46 ± 14.846 nm under the same experimental condition. We infer that this good feature comes from the common-path configuration of the CPLE.
[1] http://www.itrs.net [2] C.-C. Hsu, The applications of the heterodyne interferometry, in: I. Padron (Ed.), Interferometry-Research and Applications in Science and Technology, 2012, pp. 31–64. [3] W.T. Estler, High-accuracy displacement interferometry in air, Applied Optics 24 (1985) 808–815. [4] N. Bobroff, Residual errors in laser interferometry from air turbulence and nonlinearity, Applied Optics 26 (1987) 2676–2682. [5] C.R. Steinmetz, Sub-micron position measurement and control on precision machine tools with laser interferometry, Precision Engineering 12 (1990) 12–24. [6] A. Teimel, Technology and applications of grating interferometers in highprecision measurement, Precision Engineering 14 (1992) 147–154. [7] M. Dobosz, High-resolution laser linear encoder with numerical error compensation, Optical Engineering 38 (1999) 968–973. [8] M. Nevière, E. Popov, B. Bojhkov, L. Tsonev, S. Tonchev, High-accuracy translation-rotation encoder with two gratings in a Littrow mount, Applied Optics 38 (1999) 67–76. [9] J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, C.-C. Wu, Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution, Sensors and Actuators A 137 (2007) 185–191. [10] C.-C. Wu, W.-J. Wu, Z.-S. Pan, C.-K. Lee, Laser linear encoder with both high fabrication and head-to-scale tolerances, Applied Optics 46 (2007) 3169–3176. [11] C.F. Kao, S.H. Lu, H.M. Shen, K.C. Fan, Diffractive laser encoder with a grating in Littrow configuration, Japanese Journal of Applied Physics 47 (2008) 1833–1837. [12] J.-Y. Lee, M.-P. Lu, Optical heterodyne grating shearing interferometry for longrange positioning applications, Optics Communications 284 (2011) 857–862. [13] H.L. Hsieh, et al., Quasi-common-optical-path heterodyne grating interferometer for displacement measurement, Measurement Science and Technology 21 (2010) 115304. [14] E.G. Loewen, E. Popov, Diffraction Gratings and Applications, Marcel Dekker, New York, 1997. [15] C.-M. Wu, R.D. Deslattes, Analytical modeling of the periodic nonlinearity in heterodyne interferometry, Applied Optics 37 (1998) 6696–6700. [16] P.L.M. Heydemann, Determination and correction of quadrature fringe measurement errors in interferometers, Applied Optics 20 (1981) 3382–3384. [17] P. Gregorcic, T. Pozar, J. Mozina, Quadrature phase-shift error analysis using a homodyne laser interferometer, Optics Express 17 (2009) 16322–16331. [18] C.-M. Wu, Heterodyne interferometric system with subnanometer accuracy for measurement of straightness, Applied Optics 43 (2004) 3812–3816.
5. Conclusion
Biographies
measurement for a few meters. In Table 2, only the 4 nm pitch tolerance is allowable in order to achieve 0.1 m accuracy for 20 mm range case. According to Table 2, no grating scale is available for 30, 45, and 100 cm cases using state-of-the-art manufacturing technology. A solution is to use the double-pass configuration to increase the measurement sensitivity. This method increases the pitch tolerance. 4.3. Resolution
This study presents a novel CPLE for the measurement of nanoscale displacement. The CPLE can simultaneously achieve high environmental stability and a high resolution. It uses a twoaperture phase-shifting technique to form the quadrature signals. There are no significant discrepancies between the results of the CPLE and a commercial instrument in long-range measurements. Experimental results show that the CPLE can measure displacement down to the subnanometer range. The estimated measurement resolution is 0.1 ± 0.046 nm. Therefore, the CPLE can be applied to sub-nanometer positioning.
Chyan-Chyi Wu received his PhD degree in mechanical engineering at the National Taiwan University of Taiwan in 2001. From 2002 to 2008 he was with Industrial Technology Research Institute of Taiwan as a researcher. From 2008 till now he is with Department of Mechanical and Electromechanical Engineering in Tamkang University. His research interests are diffractive optics, nanometrology, microsensors and microactuators. Jhih-Sheng Yang received his BS degree in Huafan University in 2010. He is now working toward a MS degree at Institute of Mechanical and Electromechanical Engineering of Tamkang University. His current research activity is optical metrology and optomechatronics.
Acknowledgments
Chun-Yao Cheng received his bachelor and master degrees in aeronautics and astronautics, and mechanical engineering both at Tamkang University in 2009 and 2011 respectively. His research activity focuses on the laser encoder technology. Now he is with Compal Technology Company in Taiwan as a RD engineer.
The authors would like to acknowledge the support of the National Science Council (NSC-98-2218-E-032-009-, NSC
Yan-Zou Chen received his bachelor degree in mechanical engineering at Tamkang University in 2012. His research activity is optomechatronic system design and fabrication.
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Common-path laser encoder Chyan-Chyi Wu ∗ , Jhih-Sheng Yang, Chun-Yao Cheng, Yan-Zou Chen Advanced Photonics Laboratory, Department of Mechanical and Electro-mechanical Engineering, Tamkang University, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 21 June 2012 Received in revised form 24 August 2012 Accepted 25 August 2012 Available online 5 September 2012 Keywords: Common-path Grating Displacement Long-range Nanometer
a b s t r a c t This study proposes a novel common-path laser encoder (CPLE) capable of effectively minimizing environmental disturbance. The proposed CPLE uses a two-aperture phase-shifting technique to form quadrature signals. Experimental results match well with HP5529A results for long-range measurements. Results also show that the estimated measurement resolution is 0.1 ± 0.046 nm. Therefore, the proposed design has great potential for nanometer resolution and long-range applications. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Displacement measurements play an important role in the precise positioning systems and equipment of semiconductor and liquid crystal display (LCD) manufacturing. With the rapid advance of nanotechnology, 32 nm node on a 450 mm wafer should be feasible in 2012 [1]. Subnanometer resolution and superior immunity to environmental disturbance are required to facilitate displacement measurements over such a long range. Researchers have developed many displacement measurement techniques over the past few years. Optical interferometry is the most important of the various displacement measurement techniques. Optical interferometers are widely used for the precise measurement of displacement and other physical parameters because they have the capability of a long range and high resolution [2]. However, variations in temperature, pressure, humidity and vibration caused by environmental disturbance reduce measurement accuracy. Consequently, environmental conditions must be strictly controlled to achieve high measurement accuracy. The wavelength of the laser source also affects measurement accuracy [3–5]. Unlike optical interferometers, grating interferometers or laser encoders are not subject to environmental disturbances. These devices have better immunity to the environmental disturbances than optical interferometers because they transfer the measurement scale from the laser wavelength into the grating pitch. Teimel [6] introduced several laser encoders for submicron applications,
∗ Corresponding author at: Department of Mechanical and Electromechanical Engineering, Tamkang University, Tamsui Dist., New Taipei City 25137, Taiwan, ROC. E-mail addresses: [email protected], [email protected] (C.-C. Wu). 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.08.034
reaching a measurement resolution of 5 nm. Dobosz [7] demonstrated a laser encoder with a resolution of 10 nm. He also proposed a method to slash the optics nonlinearity of laser encoders. Nevière et al. [8] used two gratings to detect linear displacement, and the laser encoder had a measurement resolution of 80 nm. Lee et al. [9] developed an optical heterodyne laser encoder with a transmissive grating. This laser encoder is robust to environmental disturbances. Wu et al. [10] adopted a laser encoder with a perpendicular optical configuration. This laser encoder has high head-to-scale tolerance and an accuracy of 37.3 nm. Kao et al. [11] presented a laser encoder with Littrow configuration grating. This encoder achieves a maximum measurement error of 53 nm and repeatability within ±20 nm for a 100 m measurement range. Although these laser encoders achieve nanometric resolution, their optical configurations remain in a non-common path. Lee et al. [12,13] presented a quasi-common optical path laser encoder with a heterodyne laser source. They adopted the configuration of polarization interferometry, and used a half-waveplate to extract the p-polarization light from the spolarization light before modulating the measured quantity into the laser source. In this design, the p- and s-polarization light traverse side by side, but not along the same path. This device achieves a positioning resolution of 2.3 nm over a 20 mm range. However, no researcher has yet developed a common-path laser encoder. Lee’s work would be state of the art for a laser encoder because its configuration is closest to the common-path interferometer. The main problem with a common-path configuration for Lee’s laser encoder is how to perform phase shifting in the same optical path, especially after the reference and test beams interfere. This study presents a CPLE with a simple optical configuration. Because the CPLE is common-path, it possesses high measurement resolution and immunity to environmental disturbances. The CPLE
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Fig. 1. Optical configuration of the CPLE and experimental setup.
adopts a two-aperture phase-shifting technique to achieve phase shifting after the reference beam interferes with the test beam in the same optical path. This study also presents a detailed description of the CPLE operating principle. Experimental results show that the CPLE has a resolution of 0.1 ± 0.046 nm. These results demonstrate that the CPLE can measure short and long displacement with nanometric resolution. 2. Principle Fig. 1 shows the optical configuration of the CPLE. A laser source (LS) is focused by lens L1 onto the diffraction grating G and then split into 0th order and −1st order diffraction beams. These two diffracted beams diverge and overlap partly in the adjacent regions. These two divergent beams are collimated by lens L2 and then pass through the aperture stop As to extract the overlapping region of the 0th and −1st order diffracted beams (see the inset of Fig. 1). After As , these two collimated beams produce typical interference fringe. These two beams subsequently enter the beam splitter BS and pass through apertures A1 and A2 , respectively. The apertures A1 and A2 can assign an additional phase shifting for the interference signals in the CPLE or interferometers with a common path. Finally, photodetectors PD1 and PD2 measure two-beam interference fringes. This section introduces the Doppler frequency shift, which is modulated into the diffracted beams when the diffraction grating G is moving. This section also presents the principle of phase shifting using two apertures and derives the relationship between the phase variation and the grating displacement. 2.1. Doppler frequency shift in focus/divergent beam The displacement of diffraction grating G modulates a specific Doppler frequency shift ωm into each diffracted beam. This relationship can be written as [10]. ωm = vG · (k m − k i ),
(1)
where m is the diffraction order, vG is the velocity of diffraction grating G, k m is the wavevector of the mth diffraction order beam, and k i is the wavevector of the incident beam. In (1), the Doppler frequency shift depends only on the difference of the horizontal projections of the wavevectors between the incident and diffracted beams for a given velocity vG of diffraction grating G. For convenience, consider the in-plane diffraction cases shown in Fig. 2. Because the incident beam focuses onto diffraction grating G, there are separate wavevectors for various parts of the incident beam. Similarly, there are also separate wavevectors for the various parts of the specific diffraction order (divergent) beam. Consider that the focus and divergent beams comprise many finite beams, as Fig. 2 shows. The grating equation must hold for every finite beam pair in
Fig. 2. Finite beam model and schematics of different horizontal projections of wave vectors.
the focus or divergent beams. The grating equation, in the momentum conservation form, can be written as follows:
2m k m − k = , i
(2)
where is the grating pitch. Eq. (2) shows that the difference between the horizontal projections of the incident wavevector and diffracted wavevector is equal to the corresponding grating momentum, 2m/. Eq. (2) also holds for the linear grating diffraction in three dimensions. Combining (1) and (2), the Doppler frequency shift can be rewritten as follows: ωm = sign(vG )
2m |vG |,
(3)
where the sign function, sign(vG ), is equal to +1 when vG is along +X, and sign(vG ) is equal to −1 when vG is along −X. Eq. (3) shows that every finite beam pair has the same Doppler frequency shift for a specific diffraction order, irrespective of whether the beam is focused or divergent. 2.2. Interference condition for diffracted beams The proposed CPLE uses the divergent 0th and −1st order diffracted beams to interfere with each other. However, a certain condition must be met to successfully obtain the interference fringe. Consider the schematic of the overlap region of the 0th and −1st diffracted beams (Fig. 3). The grating equation reads [14] sin d = sin i +
m ,
(4)
where d is the diffraction angle of the mth order diffracted beam, i is the incident angle of the laser beam, m is the diffraction order, is the grating pitch, and is the wavelength of the laser source. Eq. (4) identifies the key diffracted angle of each marginal ray of the focusing laser beam; that is, 0 and u−1 . Fig. 3 shows that u−1 must be greater than 0 to achieve a proper overlap region between the 0th
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Fig. 3. The schematic of the overlap region of the 0th and 1st diffracted beams, where represents the convergent angle of the focusing laser beam, l represents the lower incident angle of the marginal rays, and u represents the upper incident angle of the marginal rays.
Fig. 5. A1 and A2 are aligned with the direction of the interference fringe. The term a is the width of A1 and A2 , and is the relative distance between A1 and A2 in interference fringe space.
and −1st order diffracted beams. That is, the following inequality must be met:
√ where A0 is the amplitude of E0 , and j is equal to −1. Likewise, the , can be written electric field of the −1st-order diffracted beam, E−1 as follows:
sin−1 sin( + ) −
− > 0.
(5)
Here 0 = 1 is substituted into (5), which can then be used to estimate the overlap angle of the 0th- and −1st-order diffracted beams. Fig. 4 shows the simulation results for three convergent angles of the focusing laser beam: 25◦ , 30◦ , and 50◦ . Fig. 4 shows that a large convergent angle (i.e., a lens with large numerical aperture) is helpful for the overlap of the diffracted beams. Conversely, a large angle of incidence (similar with ) is not suitable for overlap of the diffracted beams. The minimum convergent angle to achieve an overlap region of the diffracted beams is approximately 25◦ . If the 0th- and +1st-order diffracted beams interfere, the overlap angle is slightly larger than choosing the 0th- and −1st-order diffracted beams. Because the difference between these two cases is small, the proposed CPLE uses the 0th- and −1st-order diffracted beams for experimental convenience.
E−1 = A−1 ejω−1 t
1 0
,
(7)
where ω−1 represents the Doppler frequency shift of the −1st-order . These two fields, diffracted beam, and A−1 is the amplitude of E−1 , interfere with each other, and their interference fringe E0 and E−1 can be written as follows: I = A20 + A2−1 + 2A0 A−1 cos
2 q
x − ω−1 t ,
(8)
where q is the spatial period of the fringe, and x is the local coordinate for the interference fringe. Both A1 and A2 are aligned with the direction of the interference fringe (Fig. 5). By integrating (8) over the ranges (0, a) and ( , a + ), respectively, we can represent the output signals I1 and I2 of the detectors PD1 and PD2 as follows: I1 ∝ cos
a q
− ω−1 t ,
(a + 2 )
(9)
2.3. Phase shifting using two apertures
I2 ∝ cos
For the proposed CPLE (Fig. 1), the electric field of the 0th-order diffracted beam after passing L2, E0 , can be represented as follows:
where a is the width of A1 and A2 , and is the relative distance between A1 and A2 in the interference fringe space. If 2/q = (2n − 1)/2 with n belonging to the natural number in (10), we can obtain I1 and I2 in quadrature. The condition 2/q = (2n − 1)/2 can be achieved easily by changing according to the Lissajous trace of I1 and I2 signals. The common phase term in (9) and (10), a/q, can be removed in the process of phase unwrapping and the displacement calculation because the CPLE is an incremental type laser encoder. Thus, the CPLE realizes the phase shifting after the reference beam interferes with the test beam for common-path interferometers. Combining (3), (9) and (10) shows the relationship between the grating displacement X and the measured phase change ˚ as the following expression:
E0 = A0
1 0
,
(6)
X =
q
− ω−1 t ,
˚. 2
(10)
(11)
Eq. (11) shows that the grating displacement depends on the grating pitch and phase change. This equation also shows that the grating displaces a pitch distance for a full cycle of the quadrature signal. 3. Experiments and results
Fig. 4. The simulation results for three convergent angles of the focusing laser beam: 25◦ , 30◦ , and 50◦ .
Fig. 1 shows the experimental setup in this study. A frequencystabilized He–Ne laser source (model HRS015, Thorlabs, Inc.) with a wavelength of 632.991 nm (in vacuum) was used. A grating
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Fig. 6. The measured results for the stability.
Fig. 7. Measurement results of 10.8 mm displacement at different stage velocities.
with a grating pitch of 1.6 m (model: THOR-GT50-06V, Thorlabs, Inc.) was mounted on a homemade composite stage (CS), and this grating diffracted the laser source. The substrate of the grating is borosilicate glass with a thermal expansion coefficient of 3.3 × 10−6 /◦ C. To provide long- and small-range displacements, the composite stage CS was made of two types of linear stages: a longranged motorized linear stage (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 optical layout was placed on an isolated optical table (model: RS4000-412-12, Newport Inc.) to reduce the low-frequency mechanical noise from the foundation. The experiments were divided into three parts: (i) stability measurement; (ii) long-range measurement; and (iii) small-range measurement. The CPLE was used to measure the displacement of the grating or the composite-stage CS. The CPLE results were then compared with those of a commercial laser interferometer (HP5529A). The laboratory atmosphere conditions were monitored using the HP5529A sensors. All experiments were implemented under the same temperature control condition. The average atmosphere temperature was 25.4 ◦ C, with a standard deviation of 0.32 ◦ C. The average atmospheric pressure was 761.6 mmHg, with a standard deviation of 0.6 mmHg. With the thermal expansion coefficient 3.3 × 10−6 /◦ C given, for the worst case the temperature changes 2 ◦ C, the grating pitch (1.6 m) will elongate about 1.1 pm. And the effect of the grating pitch thermal expansion may linearly increase with the increase of the measured range. As our experimental results are for the travel range smaller than 10.8 mm, the error due to thermal expansion of the grating pitch will be smaller than 8 nm.
11 nm and 250 nm variations, respectively. Fig. 6 shows that the CPLE is much less sensitive to the surrounding disturbance than the HP5529A. This figure also explains why an optical interferometer is an improper candidate for nanoscale displacement measurements in engineering environments. The CPLE provides superior system stability and can efficiently reduce the influence of environmental noises.
3.1. Stability measurement Immunity to environmental disturbances can dramatically influence the applications of the proposed laser encoder. To verify the stability of the CPLE, which reflects its immunity to environmental disturbance, the grating and composite stage were held stationary in laboratory coordinates for 2 h (7200 s). We used HP5529A to monitor the stationary grating simultaneously. The experimental setup was the same as that in Fig. 1. Fig. 6 shows the measured results for stability. This figure enables estimating the long-term stability of the CPLE and HP5529A. The CPLE data truly reflect the state of the still stage. Over 2 h, there were approximately
3.2. Long-range measurement First, the target stage displacement was set at 10.8 mm based on the reading of the HP5529A, and the CPLE was used to measure the stage displacement. The sampling rate of the HP5529A was 10 Hz. Fig. 7 shows the measurement results of 10.8 mm displacement at different stage velocities. The dashed line in this figure was deliberately biased by 200 s on the time axis to differentiate between the CPLE and HP5529A data. To investigate the measurement repeatability of the CPLE for long-range measurements, we repeated the experiment of 10.8 mm displacement five times, with a stage velocity of 6.1 m/s. Table 1 shows the displacements measured by the CPLE and HP5529A for the 10.8 mm range. The measured results of the HP5529A and CPLE exhibit a few differences. The overall discrepancy was approximately 614 nm for five measurements, and the relative deviation was approximately 0.057/1000. In these experiments, the cosine error was estimated at 0.038/1000, that is, its contribution was approximately 411 nm if the included angle for the cosine error is 0.5◦ , which is frequently encountered. According to Fig. 6, with a measurement time of 1800 s, the measurement deviation between CPLE and HP5529A was approximately 100 nm. The left deviation may partly come from the cyclic nonlinearity error. According to Wu’s analysis, the typical cyclic error is about a few nanometers [15]. The manufacture tolerance of the grating may contributes partly to the measurement deviation. Generally, the grating tolerance allocates 10% measurement accuracy for a Table 1 The displacements measured by the CPLE and HP5529A for the 10.8 mm range. No.
CPLE (m)
HP5529A (m)
Deviation (nm)
Deviation ratio (%)
1 2 3 4 5
10799.615 10799.766 10799.665 10799.767 10746.912
10800.120 10800.600 10800.567 10800.364 10800.146
505 834 902 597 234
0.0047 0.0077 0.0084 0.0055 0.0022
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Fig. 8. The CPLE measurement results for the periodic triangular and sinusoidal input waveforms: (a) 5-nm range displacement and (b) 0.5-nm range displacement.
laser encoder. If an overall accuracy of 0.1 m is considered, the deviation from the grating tolerance will be 10 nm. From above analyses, we might reasonably infer that the major factor for such a discrepancy (614 nm) is caused by the cosine error (411 nm conservatively). Fortunately, better alignment can remove the cosine error. These results demonstrate that the CPLE can measure longrange displacements.
0.5 Hz and displacement amplitudes of 5 nm and 0.5 nm, respectively. Fig. 8 shows the CPLE measurement results for the periodic triangular and sinusoidal input waveforms: (i) a 5-nm range displacement and (ii) a 0.5-nm range displacement. These small-range test results indicate that the CPLE has the capability of measuring subnanometer displacements.
3.3. Small-range measurement 4. Discussion The small-range experiments in this study verified the CPLE measurement capability on the nanometer scale, especially for a displacement range of less than 10 nm. The tested ranges included 5 nm and 0.5 nm. These precise displacements were produced by moving the small-range piezoelectric stage (model: Tritor 100SG, Piezosystem Jena) of the composite stage while holding the longrange stage still. The experimental setup was as that in Fig. 1 but did not involve using HP5529A for two reasons: (i) its resolution is worse than 10 nm and (ii) its drift is too large (Fig. 6). The piezoelectric stage was driven using periodic input voltages in triangular and sinusoidal waveforms with a frequency of
Laser encoders generally suffer from two types of errors: optical path-dependent errors and component-dependent errors. Optical path-dependent errors are from misalignment (cosine and Abbe errors), vibration, temperature variation, and air turbulence. The component-dependent errors are from grating pitch tolerance, noise, and optics nonlinearity caused by polarization crosstalk. Because the CPLE is fully common-path, it essentially removes optical path-dependent errors. Because the CPLE has no polarization component, it has no optics nonlinearity error from polarization mixing. Because of its common-path configuration, the laser source
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Fig. 10. The corresponding grating pitch error is dsi for an arbitrary laser spot position Xi .
effect in operation behaves similarly to the random error. Thus, the grating pitch tolerance and the thermal effect can be combined in a random variable for subsequent analysis. Consider the total differentiation of (11) as follows:
Fig. 9. The effects of the noise level and photodetector gains on the phase change in the displacement measurement.
of the CPLE can be replaced using a short coherent length laser source, such as a compact laser diode. 4.1. Quadrature error Laser interferometers and laser encoders frequently adopt quadrature signals. The major quadrature measurement error sources are incorrect phase shift, zero offset, and gain difference [16,17]. When the phase shift between the detected signals deviates from the ideal 90◦ , the phase-unwrapping procedure generates an error in the displacement measurement. For the CPLE, the deviated interference output signals, I1 and I2 , can be expressed as I1
= (DC1 + DC1 ) + (AC1 + AC1 ) cos(˚)
(12)
and I2 = (DC2 + DC2 ) + (AC2 + AC2 ) cos(˚ + a ),
(13)
where (DCi + DCi ) and (ACi + ACi ) respectively denote the direct current and alternating current terms of the deviated intensity Ii with i = 1, 2, and a being the phase shift error between the apertures A1 and A2 . ACi causes high-frequency noises, which are mainly from the electrics. The phase shift error a in (12) and (13) can be minimized through precise alignment, and a fourphotodetector configuration can effectively remove the DC terms. Fig. 9 shows that a higher noise level spoils the measurement resolution of the CPLE. The photodetector gains also contribute periodic error to the displacement measurement. The periodic error from the photodetector gains can be reduced by fine tuning the gain of the amplification circuits. In summary, all factors of the quadrature error can be minimized through simple alignment or fine tuning the CPLE. Thus, the common periodic errors in heterodyne or homodyne laser interferometers vanish in the CPLE. 4.2. Grating pitch tolerance for different ranges The grating pitch is the measurement scale of the laser encoder. A uniform grating pitch is assumed when calculating the phase shift. However, manufacturing tolerances, incorrect installation, and the thermal effect in operation can damage the grating pitch. Thus, the grating pitch tolerance contributes to the displacement error. Manufacturing tolerances have two error sources: systematic error and random error. Systematic errors can be effectively removed using the calibration technique. Physically, the thermal
d(X) =
˚ d + d(˚). 2 2
(14)
The first term on the right side of (14) is the displacement error from the grating pitch tolerance, and the second term is the displacement error from the phase change error. The phase change error is from many factors, including optomechanical misalignment, high-frequency electric noise, optical nonlinearity, environmental disturbances, and unwrapping algorithm. Many articles have discussed these factors [9,10,17,18]. Therefore, this study concentrates on the contribution of the grating pitch tolerance. Assume an ideal case in which the phase change is error-free, that is, d(˚) = 0. Eq. (14) can be reduced as d(X) =
˚ d 2
if d(˚) = 0.
(15)
Eq. (15) shows that the displacement error is linearly proportional to the grating pitch error. However, when investigating a longrange measurement, such as 1 m, and regarding d as a constant, (15) produces an unreasonable outcome. For example, if the grating pitch = 1.6 m and the grating scale length is 1 m, then the laser spot size on the grating scale is 2 mm in diameter. A laser spot on the grating scale covers N grating lines, and N is approximately 1250. Assume that the manufacturing tolerance of the grating pitch is di for an arbitrary laser spot position Xi . Assume that the grating scale moves pitch by pitch, or cycle by cycle in interference signals. The corresponding displacement error over a pitch displacement from Xi is dsi , as shown in Fig. 10. Based on the average effect of N grating lines for a laser spot, dsi can be expressed as d dsi = √ i . N
(16)
Assume that the full measurement range is L. Within the range L, there are r times of a single pitch movement. In other words, r = [L/] + 1, where [ ] expresses the Gauss sign. The total displacement error ds can be estimated using the expression
r di 2 . ds = √ i=1
N
(17)
According to the equal accuracy principle, each pitch movement contributes an equal displacement error (i.e., ds1 = ds2 = ...). Thus, we can estimate the grating pitch error or grating pitch tolerance for different accuracies and ranges. Table 2 shows the grating pitch tolerance estimates for different measurement ranges: 30 cm, 45 cm, and 100 cm. This study uses a common commercial laser interferometer for comparison because its accuracy is a 0.1 m
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Table 2 The grating pitch tolerance estimates for different measurement ranges: 30 cm, 45 cm, and 100 cm. Measurement range (mm) 1000 450 300 200
Required accuracy (m)
Pitch tolerance (nm)
0.1 0.1 0.1 0.1
0.5 0.6 0.9 4
99-2628-E-032-001- and NSC 100-2628-E-032-001-). We also thank Professor C.K. Lee (National Taiwan University) for providing us with access to the necessary equipments. References
Because the proposed CPLE uses a 16-bit analog-to-digital converter (ADC) for displacement measurement, the theoretical displacement measurement resolution is approximately 24 pm at a grating pitch of 1.6 m. Consider the experiment shown in Fig. 1 and, when holding the stage still (i.e., d˚ = 0), the effect of the grating pitch variation can be dropped from (14). Fig. 6 shows only the phase noises. The measurement time for small-range experiments in this study is smaller than 10 s. Thus we determine the maximum and minimum data for every 10-s time interval in Fig. 6. The measurement resolution can be estimated using the difference between the maximum and minimum data. Through such a review of every 10-s time interval in Fig. 6, we can statistically obtain an estimated measurement resolution of 0.1 ± 0.046 nm. If the effective bit number for ADC is considered, the equivalent effective bit number of approximately 12 bits can be obtained. In contrast with the HP5529A, its resolution is estimated about 46 ± 14.846 nm under the same experimental condition. We infer that this good feature comes from the common-path configuration of the CPLE.
[1] http://www.itrs.net [2] C.-C. Hsu, The applications of the heterodyne interferometry, in: I. Padron (Ed.), Interferometry-Research and Applications in Science and Technology, 2012, pp. 31–64. [3] W.T. Estler, High-accuracy displacement interferometry in air, Applied Optics 24 (1985) 808–815. [4] N. Bobroff, Residual errors in laser interferometry from air turbulence and nonlinearity, Applied Optics 26 (1987) 2676–2682. [5] C.R. Steinmetz, Sub-micron position measurement and control on precision machine tools with laser interferometry, Precision Engineering 12 (1990) 12–24. [6] A. Teimel, Technology and applications of grating interferometers in highprecision measurement, Precision Engineering 14 (1992) 147–154. [7] M. Dobosz, High-resolution laser linear encoder with numerical error compensation, Optical Engineering 38 (1999) 968–973. [8] M. Nevière, E. Popov, B. Bojhkov, L. Tsonev, S. Tonchev, High-accuracy translation-rotation encoder with two gratings in a Littrow mount, Applied Optics 38 (1999) 67–76. [9] J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, C.-C. Wu, Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution, Sensors and Actuators A 137 (2007) 185–191. [10] C.-C. Wu, W.-J. Wu, Z.-S. Pan, C.-K. Lee, Laser linear encoder with both high fabrication and head-to-scale tolerances, Applied Optics 46 (2007) 3169–3176. [11] C.F. Kao, S.H. Lu, H.M. Shen, K.C. Fan, Diffractive laser encoder with a grating in Littrow configuration, Japanese Journal of Applied Physics 47 (2008) 1833–1837. [12] J.-Y. Lee, M.-P. Lu, Optical heterodyne grating shearing interferometry for longrange positioning applications, Optics Communications 284 (2011) 857–862. [13] H.L. Hsieh, et al., Quasi-common-optical-path heterodyne grating interferometer for displacement measurement, Measurement Science and Technology 21 (2010) 115304. [14] E.G. Loewen, E. Popov, Diffraction Gratings and Applications, Marcel Dekker, New York, 1997. [15] C.-M. Wu, R.D. Deslattes, Analytical modeling of the periodic nonlinearity in heterodyne interferometry, Applied Optics 37 (1998) 6696–6700. [16] P.L.M. Heydemann, Determination and correction of quadrature fringe measurement errors in interferometers, Applied Optics 20 (1981) 3382–3384. [17] P. Gregorcic, T. Pozar, J. Mozina, Quadrature phase-shift error analysis using a homodyne laser interferometer, Optics Express 17 (2009) 16322–16331. [18] C.-M. Wu, Heterodyne interferometric system with subnanometer accuracy for measurement of straightness, Applied Optics 43 (2004) 3812–3816.
5. Conclusion
Biographies
measurement for a few meters. In Table 2, only the 4 nm pitch tolerance is allowable in order to achieve 0.1 m accuracy for 20 mm range case. According to Table 2, no grating scale is available for 30, 45, and 100 cm cases using state-of-the-art manufacturing technology. A solution is to use the double-pass configuration to increase the measurement sensitivity. This method increases the pitch tolerance. 4.3. Resolution
This study presents a novel CPLE for the measurement of nanoscale displacement. The CPLE can simultaneously achieve high environmental stability and a high resolution. It uses a twoaperture phase-shifting technique to form the quadrature signals. There are no significant discrepancies between the results of the CPLE and a commercial instrument in long-range measurements. Experimental results show that the CPLE can measure displacement down to the subnanometer range. The estimated measurement resolution is 0.1 ± 0.046 nm. Therefore, the CPLE can be applied to sub-nanometer positioning.
Chyan-Chyi Wu received his PhD degree in mechanical engineering at the National Taiwan University of Taiwan in 2001. From 2002 to 2008 he was with Industrial Technology Research Institute of Taiwan as a researcher. From 2008 till now he is with Department of Mechanical and Electromechanical Engineering in Tamkang University. His research interests are diffractive optics, nanometrology, microsensors and microactuators. Jhih-Sheng Yang received his BS degree in Huafan University in 2010. He is now working toward a MS degree at Institute of Mechanical and Electromechanical Engineering of Tamkang University. His current research activity is optical metrology and optomechatronics.
Acknowledgments
Chun-Yao Cheng received his bachelor and master degrees in aeronautics and astronautics, and mechanical engineering both at Tamkang University in 2009 and 2011 respectively. His research activity focuses on the laser encoder technology. Now he is with Compal Technology Company in Taiwan as a RD engineer.
The authors would like to acknowledge the support of the National Science Council (NSC-98-2218-E-032-009-, NSC
Yan-Zou Chen received his bachelor degree in mechanical engineering at Tamkang University in 2012. His research activity is optomechatronic system design and fabrication.