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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 25553–25564
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Displacement measurement using a wavelength-phase-shifting grating interferometer

Ju-Yi Lee and Geng-An Jiang  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 25553-25564 (2013)
http://dx.doi.org/10.1364/OE.21.025553


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Abstract

A grating interferometer based on the wavelength-modulated phase-shifting method for displacement measurements is proposed. A laser beam with sequential phase shifting can be accomplished using a wavelength-modulated light passing through an unequal-path-length optical configuration. The optical phase of the moving grating is measured by the wavelength-modulated phase-shifting technique and the proposed time-domain quadrature detection method. The displacement of the grating is determined by the grating interferometry theorem with the measured phase variation. Experimental results reveal that the proposed method can detect a displacement up to a large distance of 1 mm and displacement variation down to the nanometer range.

© 2013 Optical Society of America

1. Introduction

Among the primary metrology parameters (dimension, mass, time, and frequency), the precision measurement of displacement plays an important role in modern technology. There is an increasing demand for nanometric measurement resolution in nanotechnology, semiconductors, precision manufacturing, photo-lithography, metrology instruments, high-density mass data storage systems, etc. The optical interferometer is a typical measurement tool and has been widely used for precision measurement of displacement because it offers a high measurement resolution and a wide dynamic measurement range [1

1. C. M. Wu, “Heterodyne interferometric system with sub-nanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004). [CrossRef] [PubMed]

,2

2. F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998). [CrossRef]

]. However, temperature, humidity, air pressure, and air flow in the environment must be controlled to maintain measurement accuracy [3

3. W. T. Estler, “High-accuracy displacement interferometry in air,” Appl. Opt. 24(6), 808–815 (1985). [CrossRef] [PubMed]

,4

4. M. Nevièvre, E. Popov, B. Bojhkov, L. Tsonev, and S. Tonchev, “High-accuracy translation-rotation encoder with two gratings in a Littrow mount,” Appl. Opt. 38(1), 67–76 (1999). [CrossRef] [PubMed]

]. Recently, common-optical-path heterodyne interferometers [5

5. J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011). [CrossRef]

10

10. K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009). [CrossRef]

] integrated with surface plasmon resonance (SPR) [5

5. J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011). [CrossRef]

,6

6. M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A Phys. 141(1), 217–223 (2008). [CrossRef]

] or total internal reflection (TIR) [7

7. S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009). [CrossRef] [PubMed]

,8

8. K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009). [CrossRef]

] have been developed for small displacement sensing. Their lens systems convert the displacement into an angle variation of the measurement beam. By detecting the optical phase variation of the measurement beam which passes through SPR or TIR, the displacement can be determined. Due to the common-optical-path configuration, these measurement systems can reduce environmental disturbance. However, the measurement range is only a few micrometers or less.

In contrast, the grating interferometer is independent of the light source wavelength and provides better immunity against environmental disturbances such as variations in temperature, pressure, and humidity [4

4. M. Nevièvre, E. Popov, B. Bojhkov, L. Tsonev, and S. Tonchev, “High-accuracy translation-rotation encoder with two gratings in a Littrow mount,” Appl. Opt. 38(1), 67–76 (1999). [CrossRef] [PubMed]

,11

11. A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992). [CrossRef]

14

14. J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun. 284(3), 857–862 (2011). [CrossRef]

]. Different types of grating interferometers have been developed to measure displacement with high resolution. For example, Teimel [11

11. A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992). [CrossRef]

] proposed a grating interferometer with polarization elements, and the displacement of the grating was determined by phase quadrature signals. Kao et al. [15

15. C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys. 47, 1833–1837 (2008).

] presented a diffractive laser encoder with a grating in the Littrow configuration. Kao’s laser encoder realized a maximum measurement error of 53 nm and repeatability within ± 20 nm. Wu et. al. [16

16. C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun. 297, 89–97 (2013). [CrossRef]

] designed a Littrow-type self-aligned laser encoder with double diffractions. Due to the symmetric optical configuration, Wu’s laser encoder had high tolerance. These laser encoders for grating interferometers are based on phase quadrature detection. Although these encoders have high measurement resolution, there are many optical polarization components in the phase detection system, and the optical configurations are complex. Gao et. al. [17

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

] measured the x-directional position and the z-directional out-of-straightness of a precision linear air-bearing stage with a 2-degree-of-freedom linear encoder. Recently, they further developed the multi-degree-of-freedom (DOF) surface encoder [18

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

,19

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

] for the stage motion measurement. Their multi-DOF surface encoder is composed of a planar scale grating and a reference grating which is set in the optical sensor head. The diffracted beams from the scale and reference gratings mutually interfere to generate interference signals. The multi-DOF displacements can be determined by means of analyzing the phase variations of the interference signals. Besides, the surface encoder incorporates the laser autocollimators for angular sensing. Because of the well-designed mechanical structure, Gao’s multi-DOF surface encoder is compact and has high measurement resolution. However, it is not easy to compact the optical sensor head further, because there are many optical polarization components, such as polarizers and quarter-wave plates, and photodetectors in the displacement assembly. In this paper, we proposed a novel technique for the phase quadrature detection without any polarization components. This technique can be used to reduce the size of optical sensor head.

In this study, we developed a wavelength-modulated phase-shifting method and a grating interferometer with double diffractions for displacement measurement. The principle used for this interferometry can be regarded as time-domain quadrature detection. Different from electro–optical or acousto–optical modulation, the phase shift of the light beam can be accomplished using a wavelength-modulated laser beam passing through an unequal-path-length optical configuration. We developed a new phase-extraction algorithm to calculate the optical phase variation due to the Doppler shift from the moving grating. The displacement of the grating is determined by the grating interferometry theorem with the measured phase variation. From the experimental results, the measurement range of our system is up to millimeter scale. Considering the high-frequency noise, the measurement resolution of the system is about 2 nm. The feasibility is demonstrated.

2. Principle

First, the double-diffraction interference system and the optical phase variation which results from the grating displacement are introduced in this section. Next, the wavelength-modulated technique for phase-resolution is described.

2.1 Double-diffraction interference system

E'±1exp(i2πλ2l±1±i2ϕg).
(2)

These two double-diffracted beams propagate along the same optical path and interfere with each other. The intensity of the interference detected by the photodetector is:
I|E'+1+E'1|2=1+cos(2πΔl/λ+ϕ),
(3)
where Δl = 2(l+1 − l−1) is the optical path difference of the two double-diffracted beams. ϕ = 2ϕg − (−2ϕg) = 4ϕg is the phase variation of the interference signal, which is 4 times the optical phase variation of the diffracted beams. The optical path difference Δl and the tunable wavelength of the laser diode are used to produce the phase shift for the measuring the phase variation ϕ. It is noticed that M1 and M2 can be replaced by the corner cube retro-reflectors C1 and C2 shown in Fig. 1(b). The optical configuration in Fig. 1(b) has better optical efficiency. From the above analysis, the relationship of the phase variation ϕ to the grating displacement Δx is given as:
ϕ=4ϕg=8πΔx/Λ,
(4a)
or
Δx=(Λ/8π)ϕ.
(4b)
It is obvious that the grating displacement Δx can be determined by measuring the phase variation ϕ of the interference signal.

2.2 Wavelength modulation technique and quadrature method for phase detection

In the present study, phase detection is based on the wavelength modulation technique and the quadrature method. When the LD is driven by an injection current signal S(t), the wavelength and the amplitude of the laser beam is a function of time. Considering the time-dependent injection current and the coherence length of the laser diode, the interference signal at the photodetector (Eq. (3)) can be rewritten as:
I(t)S(t)[1+Vcos(2πΔl/λ(t)+ϕ)],
(5)
where V is the visibility of the interference signal. If the driving signal is a square waveform with the period T, then the LD emits two wavelengths (λ1 and λ2) sequentially in one period. The sequential interference signal can be expressed as:
I1S1[1+Vcos(2πΔl/λ1+ϕ)],0<t<T/2,
(6a)
and
I2S2[1+Vcos(2πΔl/λ2+ϕ)],T/2<t<T,
(6b)
where S1 and S2 are the main intensities of the interference signals. Here we can select a suitable λ2 = λ1 + Δλ to make a π/2 phase difference between I2 and I1, that is:
I2S2[1+Vcos(2πλ1Δl2πλ12ΔλΔl+ϕ)]=S2[1+Vsin(2πλ1Δl+ϕ)],
(7)
where Δλ = (λ2λ1) << λ1, and 4ΔλΔl /λ12 = 1. The interference signals I1 and I2 (Eqs. (6a) and (7)) are quadrature. By adjusting the DC (S1 and S2) and AC (S1V and S2V) terms, these two signals can be used to solve the phase difference ϕ.

2.3 Selection of the wavelengths λ2 and λ1

In order to determine λ2 and λ1 for the 2 signals with π/2 phase difference, we first drive the LD with a linear increasing injection current i. Because the intensity and wavelength of the light from the LD are both proportional to the injection current, the simulated interference signal, shown graphically in Fig. 2
Fig. 2 Simulated interference signal intensity which is a function of the injection current.
, can be expressed as:
I(i)=(S0+msi)[1+Vcos(2πΔl/(λ1+mλi)+ϕ)],
(8)
where ms and mλ are the slopes of the increasing intensity and wavelength, respectively, of the laser beam which is driven by the injection current. From Fig. 2 we can find the local neighbor minimum Ia and maximum Ib at ia and ib, respectively. Of course, the phase difference between Ia and Ib is ~π. We can estimate that the phase difference π/2 will occur at iab = (ia + ib)/2, and we assume that the wavelengths λ1 and λ2 correspond to the injection currents ia and iab, respectively. Then we set the injection current at ia for wavelength λ1 and give the grating a sufficient displacement. As shown in Fig. 3
Fig. 3 The intensity of the interference signals for the injection current ia (upper curve) and wavelength λ1 (lower curve) for the injection current iab and wavelength λ2.
, the intensity of the interference signal oscillates between the minimum I1min and maximum I1max (see the upper curve in Fig. 3) because the phase ϕ increases (or decreases). The minimum I1min and maximum I1max in Eq. (6a) can be expressed as:
I1min=S1(1V),
(9a)
and
I1max=S1(1+V).
(9b)
The main intensity (or DC term) of the interference signal I1 is:
S1=(I1min+I1max)/2.
(10a)
Similarly, the main intensity S2 of the interference signal I2 (see the lower curve in Fig. 3) can be given as:
S2=(I2min+I2max)/2.
(10b)
In order to determine the phase variation ϕ, the sequential interference signals in Eqs. (6a) and (7) can be processed as:
I1=(I1S1)/S1=Vcos(2πΔl/λ1+ϕ),
(11)
and
I2=(I2S2)/S2=Vsin(2πΔl/λ1+ϕ).
(12)
The relationship between the phase variation ϕ and the modified interference signals is:

ϕ=tan1(I2/I1)2πΔl/λ1.
(13)

Equation (13) indicates that the phase variation can be determined by measuring the processed interference signals I'1 and I'2. The last term can be ignored if the optical path difference Δl is constant. Substituting the measured phase variation ϕ into Eq. (4b), the grating displacement Δx can be obtained. The curves a and b in Fig. 4
Fig. 4 Lissajous patterns of (a) the original (I1 and I2), and (b) modified (I'1 and I'2) interference signals.
show the Lissajous patterns of the original (I1 and I2) and modified (I'1 and I'2) interference signals, respectively, when the grating is given a displacement. It is well known that if the Lissajous pattern is not a circle, the calculated phase will suffer from the nonlinear error. Even though the curve b is sufficiently circular to calculate the phase ϕ, we define the error signal e to improve the Lissajous circle:
e=I12+I22.
(14)
If I'1 and I'2 have a residual DC, unequal AC terms, or the phase shift is not π/2, the error signal e will vary with the grating displacement. We can adjust the residual DC, unequal AC terms, and phase shift by tuning S1, S2, ia, and iab until the error signal e is a constant, and a much purer Lissajous circle can be obtained.

3. Performance test

3.1 Experimental setup

The configuration of this method is shown in Fig. 1(a). A laser diode (Hitachi, HL63200G) with a central wavelength of 635 nm was used as light source and modulated by a 200-Hz square-wave signal. A temperature controller was used to maintain its temperature at 15° C. A grating with a 2-μm pitch was mounted on the piezoelectric actuator, and the displacement of the linear stage was measured. The interference light was received by the photodetector PD, and the interference signal was processed by a data acquisition (DAQ) card (NI6143) and a personal computer (PC). Then the phase difference was measured using a PC-based program generated by Labview (version: 7.0, National Instruments Corporation). The phase resolution of this program is ~0.01°. All the components of the interferometer were set up on an optical table, and the room temperature was controlled at 22 °C with air conditioning. To demonstrate the feasibility of our system, we measured the displacement of a piezoelectric actuator at difference ranges. To provide long- and short-range displacements, kinds of linear stages were used, a dual-servo positioning stage (model: XYS-50; Measure control, Inc.), and a piezoelectric actuator (model: P-611; Physik Instrumente (PI) GmbH). A linear encoder with a resolution of 4 nm or strain gauge with a resolution of 1 nm was used to verify the measurement results.

3.2 Millimeter-scale displacement testing

3.3 Micrometer-scale displacement testing

The piezoelectric actuator (P-611) moves the grating in a triangular-wave form with amplitudes of 5, 10, 20, and 50 μm. These displacements are also simultaneously verified by the strain gauge sensor which is integrated on the piezoelectric actuator. Figure 6
Fig. 6 Measurement results for forward and backward displacement with amplitudes of about 50, 20, 10 and 5 μm.
shows the measurement results. The red and blue curves indicate the measurement results obtained with our method and the results obtained using the strain gauge sensor, respectively. The experimental result of strain gauges is offset by 1 s for convenient observing. From these experimental results it can be seen that displacements of different magnitudes could be measured with rather satisfactory precision.

3.4 Nanometer-scale displacement testing

To verify the measurement capability at nanometric scale, small-range experiments were done in this study. The piezoelectric actuator (P-611) drove the grating with a step-wise, and the measurement results are shown in Fig. 7
Fig. 7 Measurement results for the step-wise motion with step of 50 and 25 nm.
. There are two step sizes, 50 and 25 nm, shown in the curves. The strain gauge was also used simultaneously to confirm displacement measurements. The measurement results obtained with our method coincided well with those obtained using the strain gauge sensor.

4. Discussion

4.1 Measurement sensitivity

Based on Eq. (4), the measurement sensitivity s of our system the can be written as:
s=dϕdΔx=8πΛ.
(15)
In our experiment, we used a grating with a pitch Λ = 2 μm. From Eq. (15), a measurement sensitivity s = 0.72°/nm is obtained.

4.2 Measurement resolution and stability

Considering a phase resolution of only 0.01° in the Labview program, the measurement resolution of the displacement is about dΔx = /s ≈0.01 nm. The phase ϕ is determined by the measured intensity of the signal from Eq. (13). Therefore, the resolved intensity determines the minimum measurable phase. According to the measurement uncertainty analysis [25

25. R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci. 1(1), 3–17 (1988). [CrossRef]

], the minimum measurable phase can be written as:
dϕ=(ϕI1dI1)2+(ϕI2dI2)2=1I12+I22(I2dI1)2+(I1dI2)2.
(16)
After substituting I'1 and I'2 from Eqs. (11) and (12) into Eq. (16), can be written as:
dϕ=[(I2S2)dI1]2+[(I1S1)dI2]2S2(I1S1)2/S1+S1(I2S2)2/S2.
(17)
Here dI1 and dI2 are the minimum detectable intensity of the detector, and they have the same magnitude dI = dI1 = dI2. In our experiments, S1 is nearly equal to S2. From Eqs. (11) and (12), Eq. (17) can be simplified to:
dϕ=dISV.
(18)
Here we set the main intensity S = S1 = S2.

To test the general noise levels of the system, the piezoelectric actuator was held stationary. In this stationary situation, the contributions to the phase variation are only low- and high-frequency phase noises. The experimental results are shown in Fig. 9
Fig. 9 (a) Interference signals and (b) phase noises, including high- and low-frequency noises.
. The measurement results show that the drifting voltage of I1 and I2 was about 300 mV, and the corresponding drifting phase is about 10°. The drift of these signals may be derived from the air disturbances of non-common optical path, vibration, thermal drift. The optical path difference Δl in the last term of Eqs. (6a) and (6b) is the drift source of I1 and I2. The low frequency noises can be suppressed in the good experimental environment. Not only the low frequency noises, these curves of I1, I2 and ϕ in Fig. 9 also suffer from the high frequency noises. The high-frequency noises generated from inside of system components, such as laser source, electronic noise, photodetector, DAQ card, are inevitability. In our experimental situation, the high-frequency noises (dI) of I1, I2 are about 50 mV, the main intensity S = 3700 mV, and visibility V = 0.48. After substituting these minimum detectable intensity and parameters into Eq. (18), the minimum measurable phase = 0.028 rad (~1.6°) is obtained. From Eq. (15), the minimum measurable displacement or measurement resolution of our system is dΔx = /s ≈2 nm.

4.3 Periodic nonlinearity error analysis

The displacement measurement of our system is based on time-domain quadrature phase detection (see Eqs. (11)-(13)). Assuming that I'1 and I'2 have a residual DC, unequal AC terms, and the quadrature phase shift deviates from the ideal π/2, these two quadrature signals can be expressed as:
I1err=V1errcos(ϕ)+S1err,
(19)
and
I2err=V2errsin(ϕ+ε)+S2err,
(20)
where V1err and V2err are the unequal AC terms, S1err and S2err are the residual DC terms or biases, and ε is the phase-shifting error. Here the phase 2πΔl/λ1 is ignored. It is obvious that the Lissajous patterns of I'1err and I'2err will ellipses, and the calculated phase ϕerr will suffer from the nonlinear error δϕ:
δϕ=ϕerrϕ=tan1[V2errsin(ϕ+ε)+S2errV1errcos(ϕ)+S1err]ϕ.
(21)
According to the experimental estimation in our measurement system, the worst case for the ratio of V2err to V1err is about 1.005, and biases S1err and S2err both are about 0.005. Actually, the optical phase variation which results from the moving grating will bring about the phase-shifting error ε. That is the phase-shifting error ε is dependent on the modulation frequency f and the speed u of the grating and can be given as:
ε=8πΛu2f.
(22)
If the speed of the grating is 1 μm/s and the modulation frequency is 200 Hz, then the phase-shifting error ε is estimated to be 2°. Obviously, the higher the modulation frequency is, the smaller the phase-shifting error is. After substituting these parameters into Eq. (21), the periodic nonlinearity error can be obtained and is shown in Fig. 10
Fig. 10 Periodic nonlinearity error.
. The maximum nonlinearity phase error is about 1.5° which corresponds to 2 nm of displacement error.

4.4 Limitation of measurement speed

The measurement of the optical phase variation is based on the time-domain quadrature detection. Our PC-based program can solve successfully the continual optical phase variation, even if the wrapped phase appears in the signal. The high speed moving grating results in the high variation rate of the optical phase. If the wrapped phase rate is higher than the output data rate of the PC-based program, the missed data will cause an error of the measured phase, and limit the maximum measurement speed. According to Eq. (4), the relationship between the variation rate /dt of the optical phase and the output data rate fODR of the PC-based program can be written as
dϕdt=8πΛdΔxdt=8πΛuπfODR.
(23)
where u stands for the speed of the moving grating. The output data rate fODR of the PC-based program in our system is about 26 Hz. According to Eq. (23), the limitation of the measurement speed in our system is estimated to be 6.5 μm/s. We are developing the digital signal processor to improve the limitation of the measurement speed.

5. Conclusion

A method for displacement measurement by the wavelength phase-shifting grating interferometer with double diffraction is proposed. In our interferometer, the phase shift is accomplished by using a wavelength-modulated laser beam passing through an unequal-path-length optical configuration. We also developed a phase-extraction algorithm for time-domain quadrature detection to calculate the optical phase variation. The displacement of the grating is determined by the grating interferometry theorem with the measured phase variation. The experimental results demonstrate that the measurement resolution and range can reach nanometer and millimeter levels, respectively. Moreover, the periodic nonlinearity error caused from the residual DC, unequal AC terms of the interference signals, and the quadrature phase shift error have been discussed and analyzed.

Acknowledgments

This study was supported in part by the National Science Council, Taiwan, under contract number NSC 100-2221-E-008-074-MY3.

References and links

1.

C. M. Wu, “Heterodyne interferometric system with sub-nanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004). [CrossRef] [PubMed]

2.

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998). [CrossRef]

3.

W. T. Estler, “High-accuracy displacement interferometry in air,” Appl. Opt. 24(6), 808–815 (1985). [CrossRef] [PubMed]

4.

M. Nevièvre, E. Popov, B. Bojhkov, L. Tsonev, and S. Tonchev, “High-accuracy translation-rotation encoder with two gratings in a Littrow mount,” Appl. Opt. 38(1), 67–76 (1999). [CrossRef] [PubMed]

5.

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011). [CrossRef]

6.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A Phys. 141(1), 217–223 (2008). [CrossRef]

7.

S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009). [CrossRef] [PubMed]

8.

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009). [CrossRef]

9.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Meas. 45(6), 1510–1514 (2012). [CrossRef]

10.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009). [CrossRef]

11.

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992). [CrossRef]

12.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys. 110(1-3), 294–300 (2004). [CrossRef]

13.

C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng. 44, 023063 (2005).

14.

J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun. 284(3), 857–862 (2011). [CrossRef]

15.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys. 47, 1833–1837 (2008).

16.

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun. 297, 89–97 (2013). [CrossRef]

17.

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

18.

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

19.

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

20.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008). [CrossRef]

21.

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys. 137(1), 185–191 (2007). [CrossRef]

22.

R. Onodera and Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt. 33(22), 5052–5061 (1994). [CrossRef] [PubMed]

23.

R. Onodera and Y. Ishii, “Two-wavelength laser-diode heterodyne interferometry with one phasemeter,” Opt. Lett. 20(24), 2502–2504 (1995). [CrossRef] [PubMed]

24.

J. Y. Lee, M. P. Lu, K. Y. Lin, and S. H. Huang, “Measurement of in-plane displacement by wavelength-modulated heterodyne speckle interferometry,” Appl. Opt. 51(8), 1095–1100 (2012). [CrossRef] [PubMed]

25.

R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci. 1(1), 3–17 (1988). [CrossRef]

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

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 9, 2013
Revised Manuscript: September 27, 2013
Manuscript Accepted: October 9, 2013
Published: October 18, 2013

Citation
Ju-Yi Lee and Geng-An Jiang, "Displacement measurement using a wavelength-phase-shifting grating interferometer," Opt. Express 21, 25553-25564 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25553


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References

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  20. C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008). [CrossRef]
  21. J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007). [CrossRef]
  22. R. Onodera and Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt.33(22), 5052–5061 (1994). [CrossRef] [PubMed]
  23. R. Onodera and Y. Ishii, “Two-wavelength laser-diode heterodyne interferometry with one phasemeter,” Opt. Lett.20(24), 2502–2504 (1995). [CrossRef] [PubMed]
  24. J. Y. Lee, M. P. Lu, K. Y. Lin, and S. H. Huang, “Measurement of in-plane displacement by wavelength-modulated heterodyne speckle interferometry,” Appl. Opt.51(8), 1095–1100 (2012). [CrossRef] [PubMed]
  25. R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci.1(1), 3–17 (1988). [CrossRef]

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