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Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution
Sensors and Actuators A 137 (2007) 185–191
Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution Ju-Yi Lee a,∗ , Hui-Yi Chen a , Cheng-Chih Hsu b , Chyan-Chyi Wu b a
Institute of Opto-Mechatronics Engineering, National Central University, 300 Jhongda Rd., Jhongli City, Taoyuan County 320, Taiwan, ROC b Center for Measurement Standards, Industrial Technology Research Institute, 321 Kuang Fu Rd., Sec. 2, Hsin Chu 300, Taiwan, ROC Received 3 July 2006; received in revised form 2 February 2007; accepted 21 February 2007 Available online 24 February 2007
Abstract A novel method for displacement measurement is presented. The measurement system includes a heterodyne light source, a moving grating and a lock-in amplifier for phase measurement. The optical phase variation, which stems from the grating movement, is measured by an optical heterodyne interferometer. The experimental results demonstrate that our system can measure small and long displacement with subnanometric resolution. The theoretical resolution is about 1 pm. By means of isolating the measurement system, the low frequency noise can be reduced, and when only high frequency noises are considered, our method can achieve measurement resolution of about 0.2 nm. © 2007 Elsevier B.V. All rights reserved. Keywords: Heterodyne interferometry; Grating; Displacement; Optical phase variation
1. Introduction The precision measurement of displacement or position plays an important role in semiconductor technology, nanotechnology, biotechnology, and so on. There is an increasing demand for measuring long-range displacement with nanometric resolution. Both heterodyne [1,2] and homodyne [3,4] optical interferometers have been comprehensively used for precision measurements of displacement or other quantities which can be converted into displacement, because they offer a wide measuring range and theoretically unlimited resolution. Demarest [2] proposed a high resolution 0.3 nm heterodyne interferometer. Liu et al. [4] developed a homodyne polarization interferometer with 0.5 nm resolution. In practice, however, the optical interferometer used in non-isolation always suffers from large errors or noise, most commonly from atmospheric influences, background vibration, and thermal drift. The accuracy of the measurements in air strongly depends on the refractive index fluctuations. In order to achieve high accuracy measurement, the refractive index of air must be determined accurately by precision sensors or mathematical compensation with the Edl´en
∗
Corresponding author. Tel.: +886 3 426 7307; fax: +886 3 425 4501. E-mail address: [email protected] (J.-Y. Lee).
0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.02.017
equation [5]. Besides, to get rid of the low frequency noises, the temperature profile of the air path and the workpiece must be measured and compensated for with the help of temperature sensors. Nevertheless, the accuracy still deteriorates with increasing measured length. In contrast, grating interferometers (GI) [6–10] or optical encoders are not subject to these disturbances. In principle, the GI’s output signal is independent of light source wavelength, providing better immunity against environmental disturbances such as temperature, pressure, and humidity variation [6]. Various high-resolution displacement optical encoders have been designed using the diffraction scale principle [6–10]. For example, Teimel [9] proposed the grating interferometer with polarization elements, and the displacement of the grating was determined by the phase quadrature signals. Jourlin et al. [7,11] developed a compact displacement sensor consisting of two reflection type gratings. In the sensor, the grating period is about 1 m. By means of counting the zero crossings of the two quadrature signals, a resolution of 125 nm was obtained. Dobosz [12] used the electronic interpolator to enhance the measurement resolution to 10 nm. Kao et al. [10] proposed a similar polarization interferometry with two-dimensional grating for two-dimensional displacement measurement. In Kao’s system, a crossed grating with 1.6 m pitch in both the x- and y-axis was used. An electronic interpolation with a factor of 400
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leads to a measurement resolution of 1 nm. However, subnanometric resolution is necessary for nanotechnology. In principle, the measurement resolution increases with a smaller pitch of grating or use of higher order diffracted beams. The above references indicate that the heterodyne interferometer provides a high resolution but always suffers from surrounding noises. Although the grating interferometer prevents these surrounding noises, the measurement resolution is not enough. In this paper, we combine the grating interferometer and the optical heterodyne interferometer to reduce environmental noises and enhance the measurement resolution. In our method, the optical heterodyne interferometer was built to measure the optical phase variation of diffracted beams coming from a moving grating. The grating displacement is determined by the relationship of the optical phase variation and grating pitch. If the phase resolution (0.001◦ ) of lock-in amplifier is considered, the theoretical resolution of the proposed method is about 1 pm. Benefiting from the heterodyne interferometric phase measurement, this method has the advantages of high measurement resolution and relatively straightforward operation. The feasibility is demonstrated. 2. Principle In this section, we start by explaining the optical phase variation which results from the grating displacement. Then the optical heterodyne interferometer for the measurement of the optical phase variation is introduced. Based on the measured optical phase variation and the mathematical formula which will be derived below, the grating displacement was obtained. 2.1. Optical phase variation resulting from the grating displacement
1 + cos(2π(x0 − vt)/d) , 2
Fig. 1. A sinusoidal amplitude grating diffracts the incident light.
where k = 2π/λ is the wave number, and λ is the wavelength of the incident light; z is the distance between the grating and the screen, and l ≈ z + x2 /2z is the propagation distance of the diffracted beam. The second and third terms in Eq. (2) represent the −1st and +1st order of diffraction beams, respectively. It can be seen that when the grating is displaced along x0 axis by an amount x = vt, the optical phase in the +1st and −1st orders increases and decreases, respectively, by 2πx/d. For convenience, the amplitudes of +1st and −1st orders can be written as i2πx E±1 ∝ exp ikl±1 ± = exp(ikl±1 ± iφopv ), (3) d where φopv = 2πx/d is the optical phase variation resulting from the grating displacement x, and l+1 and l−1 are the optical path of +1st and −1st order diffraction beams, respectively. It is obvious that the optical phase variation φopv appears in the optical field. The optical phase variation φopv can be obtained with heterodyne interferometry. 2.2. Heterodyne interferometry for measuring the optical phase variation
A light beam is normally incident on a sinusoidal amplitude grating which moves along the x0 -axis with speed v, as shown in Fig. 1. The transmittance function T of the grating is defined as T (x0 , t) =
where d is the grating pitch. The incident light is of course diffracted into two orders. According to Fourier optics, the Fraunhofer diffraction pattern [13,14] in the far field is seen to be ∞ −ikxx0 2 E(x) ∝ eik(z+x /2z) T (x0 , t) exp dx0 z −∞ x x 1 = eikl δ +δ + e(−i2π/d)vt λz λz d 1 x (i2π/d)vt , (2) − +δ e λz d
(1)
We have proposed the heterodyne interferometric technique for measuring the optical phase resulting from optical activity [15] or total internal reflection [16]. With similar considerations, the schematic diagram of the optical arrangement for the grating displacement measurement is designed and shown in Fig. 2. Let the light beam coming from a laser source be linearly polarized at 45◦ with respect to the x-axis. The fast axis of electro-optic modulator EOM is along the x-axis. When this linearly polarized light passes through the EOM applied the external voltage V, the phases of the horizontal (p-polarized) and vertical (s-polarized) components of the light beam increases and decreases, respectively. The phase retardation between p- and s-polarized light depends on the external voltage V. If the phase retardation is π, then the external voltage V is called the half-wave voltage (Vλ/2 ) [17]. In our system, the EOM was driven a sawtooth voltage signal with amplitude V = Vλ/2 and angular frequency ω. The phase retardation produced by the EOM can be expressed as ωt [18]. When the original linearly polarized laser beam passes through this modulated EOM, there will be an optical frequency difference ω between the p- and s-polarized components. This light beam is similar to the Zeeman laser, and regarded as the heterodyne light source. Its Jones vector [18] is
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187
Fig. 2. The schematic diagram of the displacement measurement configuration. EOM, electro-optic modulator; G, grating; M, mirror; PBS, polarization beam splitter; AN, analyzer; D, photodetector; HPI, Hewlett-Packard interferometer.
EH = EOM(ωt)Ein ∝ =
eiωt/2 e−iωt/2
eiωt/2 0
1 −iωt/2 1 e 0
◦
E+1p = AN2 (45 )PBS(0 )E+1 ∝ exp
.
◦
×
(4)
The heterodyne light is incident normally on the grating and is diffracted into +1st and −1st order beams. Both of these two beams are heterodyned and can be symbolized by E+1 and E−1 , and are given as eiωt/2 E±1 ∝ exp(ikl±1 ± iφopv )EH =exp(ikl±1 ± iφopv ) . e−iωt/2 (5) After being reflected by mirror M1 , the +1st order beam is divided by polarization beam splitter PBS into two parts, the transmitted p-polarization light and the reflected s-polarization light. Similarly, the −1st order beam is divided by PBS into the transmitted p-polarization light and the reflected s-polarization light. That is, there are four optical paths between the grating and photodetectors: (1) G → M1 → PBS → AN2 → D2 , (2) G → M1 → PBS → AN1 → D1 , (3) G → M2 → PBS → AN1 → D1 , and (4) G → M2 → PBS → AN2 → D2 . In these four optical paths, the amplitudes of light waves received by photodetectors are symbolized by E+1p , E+1s , E−1p, and E−1s , respectively. According to Jones matrix calculation, these four amplitudes are given as 1 1 0 0 ◦ ◦ E+1s = AN1 (45 )PBS(90 )E+1 ∝ 1 1 0 1 eiωt/2 ×exp(ikl+1 + iφopv ) e−iωt/2 1 −iωt = exp , (6a) + ikl+1 + iφopv 2 1
1 1
iωt + ikl+1 + iφopv 2
(6b)
,
E−1s = AN2 (45◦ )PBS(90◦ )E−1 1 −iωt , ∝ exp + ikl−1 − iφopv 2 1 and ◦
◦
E−1p = AN1 (45 )PBS(0 )E−1 ∝ exp ×
1 1
(6c)
iωt + ikl−1 − iφopv 2
(6d)
.
It is obvious that the intensities detected by photodetector D1 and D2 are I1 ∝ |E+1s + E−1p |2 =
1 + cos[ωt + k(l−1 − l+1 ) − 2φopv ] , 2 (7)
and I2 ∝ |E−1s + E+1p |2 =
1 + cos[ωt + k(l+1 − l−1 ) + 2φopv ] , 2 (8)
respectively. These two sinusoidal signals I1 and I2 are sent into a lock-in amplifier, and the phase difference φpd between I1 and I2 is given as φpd = 4φopv + 2kl =
8πΔx + 2kl , d
(9)
where l = l+ − l− is the optical path difference between G and PBS. Eq. (9) indicates the relationship between the phase difference φpd of the two sinusoidal signals and the optical phase
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variation φopv of the diffracted beam. Assuming the optical path difference l is constant for the duration of the grating movement, the displacement s can be rewritten as x =
dφpd . 8π
(10)
It is obvious from Eq. (10) that the grating displacement x can be determined with the measurement of the phase difference φpd of the two sinusoidal signals and the known grating pitch d. 3. Experimental results To show the feasibility of our method, the displacement provided by a linear stage was measured. The experimental configuration is shown in Fig. 2. A He–Ne laser with wavelength 632.8 nm modulated by an electro-optic modulator (New Focus, Model 4002) was used as a heterodyne light source. The frequency difference between p- and s-polarized components was 1 kHz. A grating with 1.6 m pitch was mounted on the linear stage, and this grating diffracted the heterodyne light source. In order to provide long and short-range displacements, two kinds of linear stages were used in this experiment. One is the piezo-driven motorized actuator (New Focus 8095), and the other is the multilayer piezoelectric actuator (NEC/TOKIN AE0203D08). The phase variation φpd was measured by lock-in amplifier (SRS850) with an angular resolution of 0.001◦ . We set the sampling rate of the lock-in amplifier at 128 Hz in our experiment. The following two experiments show the feasibility for long and small displacement measurements.
Fig. 3. Measurement result for long displacement about 55 m. For convenient observing, the dashed line is biased 10 s.
each measurement, and due to the uncertainty of the grating pitch (d) each TD difference was not equal. Notwithstanding that a difference exists in measured results between HPI and our system, the overall discrepancy is about 0.4 m in five measurements. We suspect that the discrepancy may result from the fact that the test beam of HPI might not exactly coincide with the moving direction of grating or Abbe error, or is otherwise related to misalignment. It is possible to further improve the difference by optimizing opto-mechanical design and minimizing the alignment errors. These results demonstrate that our system is able to measure long displacements.
3.1. Long displacement 3.2. Small displacement The manual of the motorized actuator (New Focus 8095) indicates that the stage is a friction mechanism, and does not produce identical steps for each input pulse signal. Although the step size can vary slightly from pulse to pulse, it will always be less than 30 nm. According to our experience, the higher the driving speed (pulses/s), the larger the step size. The piezo-driven motorized actuator provided a long displacement (∼55 m) with a speed of 50 pulses/s, and the measured displacements are shown in Fig. 3 (solid line). A Hewlett-Packard interferometer (HPI 5529A) was used simultaneously to verify the calculated displacement (dashed line). In Fig. 3, we plot the dashed line with 10 s bias for convenient observing. Although these two curves are coincident, the total displacements (TD) which were measured by our system and HPI are a little different. The measured TDs and their difference are shown in Table 1. The heterodyne laser beam was incident at different position of the grating for Table 1 Total displacement (TD) measured by our system and HP interferometer 1st TD HPI Our system Difference
59,344 58,922 422
Values in nanometer.
2nd TD 54,233 54,663 430
3rd TD 56,388 56,167 221
4th TD 61,440 60,991 449
5th TD 56,675 56,330 345
We drove the piezo-driven motorized actuator (New Focus 8095) three steps with low speed 2 pulses/s. The experimental result is shown as Fig. 4. The measured total displacement is about 25 nm by means of our method. The measured movement of the stage by HPI 5529A is about 30 nm. Because the measurement resolution of HPI is only about 10 nm [19], HPI could not read out the detailed motion and only gave decimal scale data. Our measured data does not seem to be stable. We think the non-common optical path configuration resulted in this drift, which will be discussed in the next section. There is an error or noise signal in HPI at 1.2 s. Because HPI’s measurement area differs from ours, we believe that the error resulted from the surrounding noise of the optical path or the electronic noise in HPI’s processing unit. We also observed the similar noise from the other experiments. Although the three steps appear in both curves in Fig. 4(a), the detailed motion can be observed only in the measured results of our method and is shown in Fig. 4(b) with the 2 nm oscillation amplitude. As mentioned above, because the piezo-driven motorized actuator is based on the friction mechanism, the overshot and underdamped oscillation occurs after each pulse signal. It is obvious that our system is able to detect the overshot and underdamped oscillation, but HPI 5529A is not. The reason is that the measurement resolution of HPI is only about 10 nm, but the resolution of our system is much higher.
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Fig. 5. Measurement result for 10 steps displacement.
Fig. 4. (a) Measurement result for three steps displacement and (b) enlarged overshot and underdamped oscillation.
The resolution analysis of our system is discussed in the next section. Next, we drove the stage 10 steps with high speed 50 pulses/s. The results are shown in Fig. 5. The total displacement is about 120 nm. For the same reason, the resolution of our system is higher and our measured displacement curve is much smoother. HPI only give a decimal scale data and the noise also appeared in HPI’s curve. We conducted further experiments in which the grating was mounted on a piezoelectric actuator and moved in a saw-wave or sinusoidal form with several different amplitudes from about 450 down to 3 nm. The piezoelectric actuator was calibrated by HPI, and its displacement is about 5.1 m under 100 V driving voltage. The measurement results are shown in Fig. 6. From these results, it can be seen that our system is able to measure small displacement with good precision. The envelope of the displacement signal suffered from a low frequency drift in the smaller amplitudes (curves A and B). We think that the noncommon optical path configuration caused the drift, which will be discussed in the next section.
Fig. 6. Measurement result for displacement amplitude of about (a) 3 nm (curve A) and 12 mn (curve B), (b) 100 nm (curve C) and 450 nm (curve D).
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4. Discussion From Eq. (10), we have the measurement sensitivity of our system: φpd 8π = . x d
(11)
In our experiment, the grating with 1.6 m pitch was used. From Eq. (11), the measurement sensitivity φpd /x = 0.9 ◦ /nm is obtained. If only considering the phase resolution (φpd = 0.001◦ ) of the lock-in amplifier (SRS850), we have the measurement resolution of grating displacement about 1 pm. However, the measured phase suffers from high and low frequency noise [4]. The high-frequency phase noise may come from the laser source, EOM, photodetectors, and the operational amplifier electronics.The low-frequency phase noise may result from slow variation of the optical path difference l , such as background vibration, thermal drift, or air disturbances. If the error-free displacement xi is subject to such noises, the measured phase difference (Eq. (9)) can be rewritten as φn =
8πxi ± (|φLn | + |φHn |), d
(12)
where φn is the measured phase subject to noises, and φLn and φHn are low and high frequency phase noises, respectively. The calculated displacement xc is given as xc =
dφn d(|φLn | + |φHn |) = xi ∓ . 8π 8π
(13)
From the above equation, the smaller the grating pitch d, the smaller the phase noise influence in our system. To test these general noise levels of the system, the stage was held stationary (xi = 0). Two gratings were used to investigate the phase noise. One has a grating pitch of 40 m and the other 1.6 m. In this stationary situation, the contributors of the measured phase variation are only low and high frequency phase noises. The measurement results show that the magnitudes of the high and low frequency noises are about 1◦ and 0.2◦ within 10 s for both grating pitches 40 and 1.6 m. After substituting these magnitudes of the noises and grating pitch (40 and 1.6 m) into Eq. (13), the displacement noises are obtained and shown in Fig. 7. It is obvious that the noise level of grating pitch 1.6 m is much less than that of grating pitch 40 m in the displacement measurement using our system. In our system, the grating with 1.6 m pitch was chosen for displacement measurement. If only high frequency noise is considered, the measurement resolution of our system is estimated to be 0.2 nm. It is well known that increasing the time constant of RC filter of the lock-in amplifier will attenuate the high frequency noise level, and we believe a compact optical configuration will further improve low frequency noise. In addition, by using the two-dimensional grating instead of one-dimensional grating in similar optical configuration, the two-dimensional displacement measurements with single interferometer can be accomplished simultaneously.
Fig. 7. The calculated displacement noise, including high and low frequency.
5. Conclusion An optical heterodyne grating interferometer for displacement measurement is presented. After combining the +1st and −1st order diffracted beams, the optical phase variation resulting from the movement of grating is measured by an optical heterodyne interferometer. The grating displacement is determined by the relationship of the measured optical phase variation and the grating pitch. The experimental results demonstrate that our system is able to measure both small and long displacement with high resolution about nanometer scale. In theoretical prediction, the measurement resolution is better than 1 pm. If only high frequency noises are considered, the measurement resolution of our system is about 0.2 nm. These findings seem to indicate that our method is better than present commercial interferometers. This method has advantages such as high sensitivity and large measurement range. It can be a useful sensor to monitor the displacement and vibration of the precision motorized stage in a wide variety of research applications. Acknowledgments The authors cordially thank Center for Measurement Standards of Industrial Technology Research Institute for their useful help. This study was supported by the National Science Council, Taiwan, under contract NSC 94-2215-E-008-013. References [1] C.W. Wu, Heterodyne interferometric system with subnanometer accuracy for measurement of straightness, Appl. Opt. 43 (2004) 3812–3816. [2] F.C. Demarest, High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics, Meas. Sci. Technol. 9 (1998) 1024–1030. [3] C.K. Lee, G.Y. Wu, C.T. Teng, W.J. Wu, C.T. Lin, W.H. Hsiao, H.C. Shih, J.S. Wang, S.C. Lin, C.C. Lin, C.F. Lee, Y.C. Lin, A high performance Doppler interferometer for advanced optical storage systems, Jpn. J. Appl. Phys. 38 (1999) 1730–1741. [4] X. Liu, W. Clegg, D.F.L. Jenkins, B. Liu, Polarization interferometer for measuring small displacement, IEEE Trans. Instrum. Meas. 50 (2001) 868–871. [5] B. Edl´en, The refractive index of air, Metrologia 2 (1966) 71–80.
J.-Y. Lee et al. / Sensors and Actuators A 137 (2007) 185–191 [6] M. Neviere, E. Popov, B. Bojhkov, L. Tsonev, S. Tonchev, High-accuracy translation-rotation encoder with two gratings in a Littrow mount, Appl. Opt. 38 (1999) 67–76. [7] S. Fourment, P. Arguel, J.L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, O. Parriaux, A silicon integrated optoelectro-mechanical displacement sensor, Sens. Actuators A 110 (2004) 294–300. [8] M. Tabib-Azar, A. Garcia-Valenzuela, Sensing means and sensor shells: a new method of comparative study of piezoelectric, piezoresistive, electrostatic, magnetic and optical sensors, Sens. Actuators A 48 (1995) 87–100. [9] A. Teimel, Technology and applications of grating interferometers in highprecision measurement, Prec. Eng. 14 (1992) 147–154. [10] C.F. Kao, C.C. Chang, M.H. Lu, Double-diffraction planar encoder by conjugate optics, Opt. Eng. 44 (2005) 023063. [11] Y. Jourlin, J. Jay, O. Parriaux, Compact diffractive interferometric displacement sensor in reflection, Prec. Eng. 26 (2002) 1–6. [12] M. Dobosz, High-resolution laser linear encoder with numerical error compensation, Opt. Eng. 38 (1999) 968–973. [13] W.H. Stevenson, Optical frequency shifting by means of rotating diffraction grating, Appl. Opt. 9 (1970) 649–652. [14] K.D. M¨oller, Optics, vol. 1, Mill-Valley, California, 1988, p. 140. [15] J.Y. Lee, D.C. Su, Improved common-path optical heterodyne interferometer for measuring small optical rotation angle of chiral medium, Opt. Commun. 256 (2005) 337–341. [16] M.H. Chiu, J.Y. Lee, D.C. Su, K.H. Lee, Vacuum measurement using total-internal-reflection heterodyne interferometry, Prec. Eng. 23 (1999) 260–263. [17] S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Prentice Hall, New Jersey, 2001, p. 298.
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[18] D.C. Su, M.H. Chiu, C.D. Chen, Simple two-frequency laser, Prec. Eng. 18 (1996) 161–163. [19] Agilent 5529A Dynamic Calibrator Datasheet.
Biographies Ju-Yi Lee received his PhD degree from the Institute of Electro-Optical Engineering, the National Chiao Tung University of Taiwan in 1999. Prior to joining the National Central University as an assistant professor (since 2004), he was a researcher in Industrial Technology Research Institute of Taiwan. His research interests are optical metrology and sensors. Hui-Yi Chen received his BS degree in industrial education at the National Taiwan Normal University in 2003. He is now working toward a MS degree at the Institute of Mechanical Engineering of National Central University of Taiwan. His current research activity is optical metrology. Cheng-Chih Hsu received his PhD degree from the Institute of Electro-Optical Engineering, the National Chiao Tung University of Taiwan in 2003. He is with Industrial Technology Research Institute of Taiwan, as a researcher. His current research activities are optical metrology, nondestructive testing and optical measurement in medical diagnostics. Chyan-Chyi Wu received his PhD degree in mechanical engineering at the National Taiwan University of Taiwan in 2001. He is with Industrial Technology Research Institute of Taiwan, as a researcher. His research interests are diffractive optics, nanometrology and microsensors and actuators.
Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution Ju-Yi Lee a,∗ , Hui-Yi Chen a , Cheng-Chih Hsu b , Chyan-Chyi Wu b a
Institute of Opto-Mechatronics Engineering, National Central University, 300 Jhongda Rd., Jhongli City, Taoyuan County 320, Taiwan, ROC b Center for Measurement Standards, Industrial Technology Research Institute, 321 Kuang Fu Rd., Sec. 2, Hsin Chu 300, Taiwan, ROC Received 3 July 2006; received in revised form 2 February 2007; accepted 21 February 2007 Available online 24 February 2007
Abstract A novel method for displacement measurement is presented. The measurement system includes a heterodyne light source, a moving grating and a lock-in amplifier for phase measurement. The optical phase variation, which stems from the grating movement, is measured by an optical heterodyne interferometer. The experimental results demonstrate that our system can measure small and long displacement with subnanometric resolution. The theoretical resolution is about 1 pm. By means of isolating the measurement system, the low frequency noise can be reduced, and when only high frequency noises are considered, our method can achieve measurement resolution of about 0.2 nm. © 2007 Elsevier B.V. All rights reserved. Keywords: Heterodyne interferometry; Grating; Displacement; Optical phase variation
1. Introduction The precision measurement of displacement or position plays an important role in semiconductor technology, nanotechnology, biotechnology, and so on. There is an increasing demand for measuring long-range displacement with nanometric resolution. Both heterodyne [1,2] and homodyne [3,4] optical interferometers have been comprehensively used for precision measurements of displacement or other quantities which can be converted into displacement, because they offer a wide measuring range and theoretically unlimited resolution. Demarest [2] proposed a high resolution 0.3 nm heterodyne interferometer. Liu et al. [4] developed a homodyne polarization interferometer with 0.5 nm resolution. In practice, however, the optical interferometer used in non-isolation always suffers from large errors or noise, most commonly from atmospheric influences, background vibration, and thermal drift. The accuracy of the measurements in air strongly depends on the refractive index fluctuations. In order to achieve high accuracy measurement, the refractive index of air must be determined accurately by precision sensors or mathematical compensation with the Edl´en
∗
Corresponding author. Tel.: +886 3 426 7307; fax: +886 3 425 4501. E-mail address: [email protected] (J.-Y. Lee).
0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.02.017
equation [5]. Besides, to get rid of the low frequency noises, the temperature profile of the air path and the workpiece must be measured and compensated for with the help of temperature sensors. Nevertheless, the accuracy still deteriorates with increasing measured length. In contrast, grating interferometers (GI) [6–10] or optical encoders are not subject to these disturbances. In principle, the GI’s output signal is independent of light source wavelength, providing better immunity against environmental disturbances such as temperature, pressure, and humidity variation [6]. Various high-resolution displacement optical encoders have been designed using the diffraction scale principle [6–10]. For example, Teimel [9] proposed the grating interferometer with polarization elements, and the displacement of the grating was determined by the phase quadrature signals. Jourlin et al. [7,11] developed a compact displacement sensor consisting of two reflection type gratings. In the sensor, the grating period is about 1 m. By means of counting the zero crossings of the two quadrature signals, a resolution of 125 nm was obtained. Dobosz [12] used the electronic interpolator to enhance the measurement resolution to 10 nm. Kao et al. [10] proposed a similar polarization interferometry with two-dimensional grating for two-dimensional displacement measurement. In Kao’s system, a crossed grating with 1.6 m pitch in both the x- and y-axis was used. An electronic interpolation with a factor of 400
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leads to a measurement resolution of 1 nm. However, subnanometric resolution is necessary for nanotechnology. In principle, the measurement resolution increases with a smaller pitch of grating or use of higher order diffracted beams. The above references indicate that the heterodyne interferometer provides a high resolution but always suffers from surrounding noises. Although the grating interferometer prevents these surrounding noises, the measurement resolution is not enough. In this paper, we combine the grating interferometer and the optical heterodyne interferometer to reduce environmental noises and enhance the measurement resolution. In our method, the optical heterodyne interferometer was built to measure the optical phase variation of diffracted beams coming from a moving grating. The grating displacement is determined by the relationship of the optical phase variation and grating pitch. If the phase resolution (0.001◦ ) of lock-in amplifier is considered, the theoretical resolution of the proposed method is about 1 pm. Benefiting from the heterodyne interferometric phase measurement, this method has the advantages of high measurement resolution and relatively straightforward operation. The feasibility is demonstrated. 2. Principle In this section, we start by explaining the optical phase variation which results from the grating displacement. Then the optical heterodyne interferometer for the measurement of the optical phase variation is introduced. Based on the measured optical phase variation and the mathematical formula which will be derived below, the grating displacement was obtained. 2.1. Optical phase variation resulting from the grating displacement
1 + cos(2π(x0 − vt)/d) , 2
Fig. 1. A sinusoidal amplitude grating diffracts the incident light.
where k = 2π/λ is the wave number, and λ is the wavelength of the incident light; z is the distance between the grating and the screen, and l ≈ z + x2 /2z is the propagation distance of the diffracted beam. The second and third terms in Eq. (2) represent the −1st and +1st order of diffraction beams, respectively. It can be seen that when the grating is displaced along x0 axis by an amount x = vt, the optical phase in the +1st and −1st orders increases and decreases, respectively, by 2πx/d. For convenience, the amplitudes of +1st and −1st orders can be written as i2πx E±1 ∝ exp ikl±1 ± = exp(ikl±1 ± iφopv ), (3) d where φopv = 2πx/d is the optical phase variation resulting from the grating displacement x, and l+1 and l−1 are the optical path of +1st and −1st order diffraction beams, respectively. It is obvious that the optical phase variation φopv appears in the optical field. The optical phase variation φopv can be obtained with heterodyne interferometry. 2.2. Heterodyne interferometry for measuring the optical phase variation
A light beam is normally incident on a sinusoidal amplitude grating which moves along the x0 -axis with speed v, as shown in Fig. 1. The transmittance function T of the grating is defined as T (x0 , t) =
where d is the grating pitch. The incident light is of course diffracted into two orders. According to Fourier optics, the Fraunhofer diffraction pattern [13,14] in the far field is seen to be ∞ −ikxx0 2 E(x) ∝ eik(z+x /2z) T (x0 , t) exp dx0 z −∞ x x 1 = eikl δ +δ + e(−i2π/d)vt λz λz d 1 x (i2π/d)vt , (2) − +δ e λz d
(1)
We have proposed the heterodyne interferometric technique for measuring the optical phase resulting from optical activity [15] or total internal reflection [16]. With similar considerations, the schematic diagram of the optical arrangement for the grating displacement measurement is designed and shown in Fig. 2. Let the light beam coming from a laser source be linearly polarized at 45◦ with respect to the x-axis. The fast axis of electro-optic modulator EOM is along the x-axis. When this linearly polarized light passes through the EOM applied the external voltage V, the phases of the horizontal (p-polarized) and vertical (s-polarized) components of the light beam increases and decreases, respectively. The phase retardation between p- and s-polarized light depends on the external voltage V. If the phase retardation is π, then the external voltage V is called the half-wave voltage (Vλ/2 ) [17]. In our system, the EOM was driven a sawtooth voltage signal with amplitude V = Vλ/2 and angular frequency ω. The phase retardation produced by the EOM can be expressed as ωt [18]. When the original linearly polarized laser beam passes through this modulated EOM, there will be an optical frequency difference ω between the p- and s-polarized components. This light beam is similar to the Zeeman laser, and regarded as the heterodyne light source. Its Jones vector [18] is
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Fig. 2. The schematic diagram of the displacement measurement configuration. EOM, electro-optic modulator; G, grating; M, mirror; PBS, polarization beam splitter; AN, analyzer; D, photodetector; HPI, Hewlett-Packard interferometer.
EH = EOM(ωt)Ein ∝ =
eiωt/2 e−iωt/2
eiωt/2 0
1 −iωt/2 1 e 0
◦
E+1p = AN2 (45 )PBS(0 )E+1 ∝ exp
.
◦
×
(4)
The heterodyne light is incident normally on the grating and is diffracted into +1st and −1st order beams. Both of these two beams are heterodyned and can be symbolized by E+1 and E−1 , and are given as eiωt/2 E±1 ∝ exp(ikl±1 ± iφopv )EH =exp(ikl±1 ± iφopv ) . e−iωt/2 (5) After being reflected by mirror M1 , the +1st order beam is divided by polarization beam splitter PBS into two parts, the transmitted p-polarization light and the reflected s-polarization light. Similarly, the −1st order beam is divided by PBS into the transmitted p-polarization light and the reflected s-polarization light. That is, there are four optical paths between the grating and photodetectors: (1) G → M1 → PBS → AN2 → D2 , (2) G → M1 → PBS → AN1 → D1 , (3) G → M2 → PBS → AN1 → D1 , and (4) G → M2 → PBS → AN2 → D2 . In these four optical paths, the amplitudes of light waves received by photodetectors are symbolized by E+1p , E+1s , E−1p, and E−1s , respectively. According to Jones matrix calculation, these four amplitudes are given as 1 1 0 0 ◦ ◦ E+1s = AN1 (45 )PBS(90 )E+1 ∝ 1 1 0 1 eiωt/2 ×exp(ikl+1 + iφopv ) e−iωt/2 1 −iωt = exp , (6a) + ikl+1 + iφopv 2 1
1 1
iωt + ikl+1 + iφopv 2
(6b)
,
E−1s = AN2 (45◦ )PBS(90◦ )E−1 1 −iωt , ∝ exp + ikl−1 − iφopv 2 1 and ◦
◦
E−1p = AN1 (45 )PBS(0 )E−1 ∝ exp ×
1 1
(6c)
iωt + ikl−1 − iφopv 2
(6d)
.
It is obvious that the intensities detected by photodetector D1 and D2 are I1 ∝ |E+1s + E−1p |2 =
1 + cos[ωt + k(l−1 − l+1 ) − 2φopv ] , 2 (7)
and I2 ∝ |E−1s + E+1p |2 =
1 + cos[ωt + k(l+1 − l−1 ) + 2φopv ] , 2 (8)
respectively. These two sinusoidal signals I1 and I2 are sent into a lock-in amplifier, and the phase difference φpd between I1 and I2 is given as φpd = 4φopv + 2kl =
8πΔx + 2kl , d
(9)
where l = l+ − l− is the optical path difference between G and PBS. Eq. (9) indicates the relationship between the phase difference φpd of the two sinusoidal signals and the optical phase
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variation φopv of the diffracted beam. Assuming the optical path difference l is constant for the duration of the grating movement, the displacement s can be rewritten as x =
dφpd . 8π
(10)
It is obvious from Eq. (10) that the grating displacement x can be determined with the measurement of the phase difference φpd of the two sinusoidal signals and the known grating pitch d. 3. Experimental results To show the feasibility of our method, the displacement provided by a linear stage was measured. The experimental configuration is shown in Fig. 2. A He–Ne laser with wavelength 632.8 nm modulated by an electro-optic modulator (New Focus, Model 4002) was used as a heterodyne light source. The frequency difference between p- and s-polarized components was 1 kHz. A grating with 1.6 m pitch was mounted on the linear stage, and this grating diffracted the heterodyne light source. In order to provide long and short-range displacements, two kinds of linear stages were used in this experiment. One is the piezo-driven motorized actuator (New Focus 8095), and the other is the multilayer piezoelectric actuator (NEC/TOKIN AE0203D08). The phase variation φpd was measured by lock-in amplifier (SRS850) with an angular resolution of 0.001◦ . We set the sampling rate of the lock-in amplifier at 128 Hz in our experiment. The following two experiments show the feasibility for long and small displacement measurements.
Fig. 3. Measurement result for long displacement about 55 m. For convenient observing, the dashed line is biased 10 s.
each measurement, and due to the uncertainty of the grating pitch (d) each TD difference was not equal. Notwithstanding that a difference exists in measured results between HPI and our system, the overall discrepancy is about 0.4 m in five measurements. We suspect that the discrepancy may result from the fact that the test beam of HPI might not exactly coincide with the moving direction of grating or Abbe error, or is otherwise related to misalignment. It is possible to further improve the difference by optimizing opto-mechanical design and minimizing the alignment errors. These results demonstrate that our system is able to measure long displacements.
3.1. Long displacement 3.2. Small displacement The manual of the motorized actuator (New Focus 8095) indicates that the stage is a friction mechanism, and does not produce identical steps for each input pulse signal. Although the step size can vary slightly from pulse to pulse, it will always be less than 30 nm. According to our experience, the higher the driving speed (pulses/s), the larger the step size. The piezo-driven motorized actuator provided a long displacement (∼55 m) with a speed of 50 pulses/s, and the measured displacements are shown in Fig. 3 (solid line). A Hewlett-Packard interferometer (HPI 5529A) was used simultaneously to verify the calculated displacement (dashed line). In Fig. 3, we plot the dashed line with 10 s bias for convenient observing. Although these two curves are coincident, the total displacements (TD) which were measured by our system and HPI are a little different. The measured TDs and their difference are shown in Table 1. The heterodyne laser beam was incident at different position of the grating for Table 1 Total displacement (TD) measured by our system and HP interferometer 1st TD HPI Our system Difference
59,344 58,922 422
Values in nanometer.
2nd TD 54,233 54,663 430
3rd TD 56,388 56,167 221
4th TD 61,440 60,991 449
5th TD 56,675 56,330 345
We drove the piezo-driven motorized actuator (New Focus 8095) three steps with low speed 2 pulses/s. The experimental result is shown as Fig. 4. The measured total displacement is about 25 nm by means of our method. The measured movement of the stage by HPI 5529A is about 30 nm. Because the measurement resolution of HPI is only about 10 nm [19], HPI could not read out the detailed motion and only gave decimal scale data. Our measured data does not seem to be stable. We think the non-common optical path configuration resulted in this drift, which will be discussed in the next section. There is an error or noise signal in HPI at 1.2 s. Because HPI’s measurement area differs from ours, we believe that the error resulted from the surrounding noise of the optical path or the electronic noise in HPI’s processing unit. We also observed the similar noise from the other experiments. Although the three steps appear in both curves in Fig. 4(a), the detailed motion can be observed only in the measured results of our method and is shown in Fig. 4(b) with the 2 nm oscillation amplitude. As mentioned above, because the piezo-driven motorized actuator is based on the friction mechanism, the overshot and underdamped oscillation occurs after each pulse signal. It is obvious that our system is able to detect the overshot and underdamped oscillation, but HPI 5529A is not. The reason is that the measurement resolution of HPI is only about 10 nm, but the resolution of our system is much higher.
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Fig. 5. Measurement result for 10 steps displacement.
Fig. 4. (a) Measurement result for three steps displacement and (b) enlarged overshot and underdamped oscillation.
The resolution analysis of our system is discussed in the next section. Next, we drove the stage 10 steps with high speed 50 pulses/s. The results are shown in Fig. 5. The total displacement is about 120 nm. For the same reason, the resolution of our system is higher and our measured displacement curve is much smoother. HPI only give a decimal scale data and the noise also appeared in HPI’s curve. We conducted further experiments in which the grating was mounted on a piezoelectric actuator and moved in a saw-wave or sinusoidal form with several different amplitudes from about 450 down to 3 nm. The piezoelectric actuator was calibrated by HPI, and its displacement is about 5.1 m under 100 V driving voltage. The measurement results are shown in Fig. 6. From these results, it can be seen that our system is able to measure small displacement with good precision. The envelope of the displacement signal suffered from a low frequency drift in the smaller amplitudes (curves A and B). We think that the noncommon optical path configuration caused the drift, which will be discussed in the next section.
Fig. 6. Measurement result for displacement amplitude of about (a) 3 nm (curve A) and 12 mn (curve B), (b) 100 nm (curve C) and 450 nm (curve D).
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4. Discussion From Eq. (10), we have the measurement sensitivity of our system: φpd 8π = . x d
(11)
In our experiment, the grating with 1.6 m pitch was used. From Eq. (11), the measurement sensitivity φpd /x = 0.9 ◦ /nm is obtained. If only considering the phase resolution (φpd = 0.001◦ ) of the lock-in amplifier (SRS850), we have the measurement resolution of grating displacement about 1 pm. However, the measured phase suffers from high and low frequency noise [4]. The high-frequency phase noise may come from the laser source, EOM, photodetectors, and the operational amplifier electronics.The low-frequency phase noise may result from slow variation of the optical path difference l , such as background vibration, thermal drift, or air disturbances. If the error-free displacement xi is subject to such noises, the measured phase difference (Eq. (9)) can be rewritten as φn =
8πxi ± (|φLn | + |φHn |), d
(12)
where φn is the measured phase subject to noises, and φLn and φHn are low and high frequency phase noises, respectively. The calculated displacement xc is given as xc =
dφn d(|φLn | + |φHn |) = xi ∓ . 8π 8π
(13)
From the above equation, the smaller the grating pitch d, the smaller the phase noise influence in our system. To test these general noise levels of the system, the stage was held stationary (xi = 0). Two gratings were used to investigate the phase noise. One has a grating pitch of 40 m and the other 1.6 m. In this stationary situation, the contributors of the measured phase variation are only low and high frequency phase noises. The measurement results show that the magnitudes of the high and low frequency noises are about 1◦ and 0.2◦ within 10 s for both grating pitches 40 and 1.6 m. After substituting these magnitudes of the noises and grating pitch (40 and 1.6 m) into Eq. (13), the displacement noises are obtained and shown in Fig. 7. It is obvious that the noise level of grating pitch 1.6 m is much less than that of grating pitch 40 m in the displacement measurement using our system. In our system, the grating with 1.6 m pitch was chosen for displacement measurement. If only high frequency noise is considered, the measurement resolution of our system is estimated to be 0.2 nm. It is well known that increasing the time constant of RC filter of the lock-in amplifier will attenuate the high frequency noise level, and we believe a compact optical configuration will further improve low frequency noise. In addition, by using the two-dimensional grating instead of one-dimensional grating in similar optical configuration, the two-dimensional displacement measurements with single interferometer can be accomplished simultaneously.
Fig. 7. The calculated displacement noise, including high and low frequency.
5. Conclusion An optical heterodyne grating interferometer for displacement measurement is presented. After combining the +1st and −1st order diffracted beams, the optical phase variation resulting from the movement of grating is measured by an optical heterodyne interferometer. The grating displacement is determined by the relationship of the measured optical phase variation and the grating pitch. The experimental results demonstrate that our system is able to measure both small and long displacement with high resolution about nanometer scale. In theoretical prediction, the measurement resolution is better than 1 pm. If only high frequency noises are considered, the measurement resolution of our system is about 0.2 nm. These findings seem to indicate that our method is better than present commercial interferometers. This method has advantages such as high sensitivity and large measurement range. It can be a useful sensor to monitor the displacement and vibration of the precision motorized stage in a wide variety of research applications. Acknowledgments The authors cordially thank Center for Measurement Standards of Industrial Technology Research Institute for their useful help. This study was supported by the National Science Council, Taiwan, under contract NSC 94-2215-E-008-013. References [1] C.W. Wu, Heterodyne interferometric system with subnanometer accuracy for measurement of straightness, Appl. Opt. 43 (2004) 3812–3816. [2] F.C. Demarest, High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics, Meas. Sci. Technol. 9 (1998) 1024–1030. [3] C.K. Lee, G.Y. Wu, C.T. Teng, W.J. Wu, C.T. Lin, W.H. Hsiao, H.C. Shih, J.S. Wang, S.C. Lin, C.C. Lin, C.F. Lee, Y.C. Lin, A high performance Doppler interferometer for advanced optical storage systems, Jpn. J. Appl. Phys. 38 (1999) 1730–1741. [4] X. Liu, W. Clegg, D.F.L. Jenkins, B. Liu, Polarization interferometer for measuring small displacement, IEEE Trans. Instrum. Meas. 50 (2001) 868–871. [5] B. Edl´en, The refractive index of air, Metrologia 2 (1966) 71–80.
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Biographies Ju-Yi Lee received his PhD degree from the Institute of Electro-Optical Engineering, the National Chiao Tung University of Taiwan in 1999. Prior to joining the National Central University as an assistant professor (since 2004), he was a researcher in Industrial Technology Research Institute of Taiwan. His research interests are optical metrology and sensors. Hui-Yi Chen received his BS degree in industrial education at the National Taiwan Normal University in 2003. He is now working toward a MS degree at the Institute of Mechanical Engineering of National Central University of Taiwan. His current research activity is optical metrology. Cheng-Chih Hsu received his PhD degree from the Institute of Electro-Optical Engineering, the National Chiao Tung University of Taiwan in 2003. He is with Industrial Technology Research Institute of Taiwan, as a researcher. His current research activities are optical metrology, nondestructive testing and optical measurement in medical diagnostics. Chyan-Chyi Wu received his PhD degree in mechanical engineering at the National Taiwan University of Taiwan in 2001. He is with Industrial Technology Research Institute of Taiwan, as a researcher. His research interests are diffractive optics, nanometrology and microsensors and actuators.