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 [1,2]. However, temperature, humidity, air pressure, and air flow in the environment must be controlled to maintain measurement accuracy [3,4]. Recently, common-optical-path heterodyne interferometers [510] integrated with surface plasmon resonance (SPR) [5,6] or total internal reflection (TIR) [7,8] 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 [4,1114]. Different types of grating interferometers have been developed to measure displacement with high resolution. For example, Teimel [11] proposed a grating interferometer with polarization elements, and the displacement of the grating was determined by phase quadrature signals. Kao et al. [15] 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] 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] 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 [18,19] 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.

Generally, heterodyne detection can effectively overcome the common problem of DC offset and amplitude variation of the homodyne grating interferometer. Hsu et. al. [20] developed a reflection-type heterodyne grating interferometer for in-plane displacement measurement with a resolution of 0.5 nm. In 2007, we proposed a heterodyne grating interferometer for measuring in-plane displacement with a resolution of 0.2 nm [21]. In the heterodyne device, an electro–optical modulator installed in the interferometer modulates the laser beams at different frequencies. Although the measurement resolution is high, the electro–optical modulator is quite expensive and bulky. Ishii et. al [22,23] conducted several studies on heterodyne interferometry with a frequency-modulated (or wavelength-modulated) laser diode for surface profile measurement. The injection current is continuously changed to introduce a time-varying phase difference between the two beams of an unbalanced Twyman–Green interferometer. By analyzing the time-varying interference fringes, the interference phase and surface profile can be determined. Following the thought of the wavelength-modulated heterodyne detection, we presented a method of wavelength-modulated heterodyne speckle interferometry for in-plane displacement measurement [24]. Different from the electro–optical-modulation and acousto–optical-modulation methods, we combined an optical-path-difference configuration with wavelength modulation of a laser diode source by injection current modulation to achieve heterodyne detection. The displacement of the object can be determined by the speckle interferometry theorem with heterodyne phase detection. This previous work also demonstrated that the measurement system can detect a displacement variation down to nanometer scale with a measurement range of hundreds of micrometers. However, not only the wavelength, but also the light intensity of the laser diode is modulated by the injection current. The modulated intensity that causes the interference signal is not a pure sinusoid curve, and phase detection is difficult and inconvenient.

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

A schematic diagram of the double-diffraction interference system is shown in Fig. 1. For convenience, the + z axis is chosen to be along the direction of propagation, and the x axis is along the horizontal direction. A beam from the laser diode passes through the beam splitter BS and is incident onto the diffraction grating G. The laser beam is diffracted into the + 1st- and −1st-order beams. According to Fourier optics analysis in our previous work [21], when the grating is displaced along x axis by an amount Δx, the optical phase in the + 1st- and −1st-order beams increases and decreases, respectively, by ϕg = 2πΔx/Λ. Here Λ is the grating pitch. For convenience, we assume that the amplitude of the original laser beam is 1, then the amplitudes (E+1, E-1) of these two diffraction beams can be written as:

E±1exp(i2πλl±1±iϕg).
Here 2π/λ is the wave number, λ is the wavelength of the laser beam, and l+1 and l−1 are the optical paths of the + 1st- and −1st-order beams from the grating to mirrors M1 and M2, respectively. Then, these two diffraction beams are reflected from M1 and M2, and diffracted again by the grating G. These two double-diffracted beams can be expressed as:

 

Fig. 1 Schematic diagram of the wavelength phase-shifting grating interferometer. The diffraction beams can be reflected by (a) the mirrors or (b) the corner cube retro-reflectors. These reflected beams are diffracted by the grating G again, and interfere with each other. FG: Function Generator, LD: Laser Diode, BS: Beam Splitter, G: Grating, M: Mirror, C: Corner Cube Retro-reflectors, PZT: Piezoelectric actuators, PD: Photodetector, PC: Personal Computer.

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E'±1exp(i2πλ2l±1±i2ϕg).

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),
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,
or
Δx=(Λ/8π)ϕ.
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)],
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,
and
I2S2[1+Vcos(2πΔl/λ2+ϕ)],T/2<t<T,
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)],
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, can be expressed as:

I(i)=(S0+msi)[1+Vcos(2πΔl/(λ1+mλi)+ϕ)],
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, 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),
and
I1max=S1(1+V).
The main intensity (or DC term) of the interference signal I1 is:
S1=(I1min+I1max)/2.
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.
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),
and
I2=(I2S2)/S2=Vsin(2πΔl/λ1+ϕ).
The relationship between the phase variation ϕ and the modified interference signals is:

 

Fig. 2 Simulated interference signal intensity which is a function of the injection current.

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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.

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ϕ=tan1(I2/I1)2πΔl/λ1.

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 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.
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.

 

Fig. 4 Lissajous patterns of (a) the original (I1 and I2), and (b) modified (I'1 and I'2) interference signals.

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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

A grating was mounted on a linear stage (XYS-50) with a 1-mm traveling range for long-range displacement testing. This stage was equipped with a linear encoder which was used to verify the movement simultaneously, and was operated in a closed-loop configuration for millimeter traveling range positioning. The stage provided a long forward displacement of ~1 mm at 2 different speeds, 2 and 3.3 μm/s. The measured displacements (red curves) are shown in Fig. 5. The curves measured by the encoder are offset by a few seconds for convenience of observation (blue curves). The measurement results obtained with the proposed method appear to coincide well with those obtained with the encoder. This indicates that our interferometer is able to measure large displacement (millimeter range) like a commercial sensor. After several measurements of the linear displacement (~1 mm) and statistical analysis, the difference in displacement measured by the encoder, and our system was less than 0.6 μm. We suspect that the discrepancy may result from the fact that the moving direction of grating might not exactly coincide with the one of the linear encoder, or is otherwise related to misalignment. By optimizing opto–mechanical design and minimizing the alignment errors, the discrepancy can be further improved.

 

Fig. 5 Measurement results for a long displacement of ~1 mm. Red curves: measured displacements, blue curves: measured by the encoder. Curves are offset by a few seconds for convenience of observation.

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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 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.

 

Fig. 6 Measurement results for forward and backward displacement with amplitudes of about 50, 20, 10 and 5 μm.

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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. 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.

 

Fig. 7 Measurement results for the step-wise motion with step of 50 and 25 nm.

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Figure 8 shows the measurement results for the 560 nm and 300 nm displacements with steps of 10 and 5 nm. The 10-nm displacement is easily observable by the strain gauge (blue curves) and our system (red curves) in the upper-left inset of Fig. 8. The lower curves and inset of Fig. 8 show the results of the 5-nm displacement testing. Despite the fact that the displacement results obtained with our method seem to suffer from low-frequency drift, the 5-nm displacement still can be observed in our system. We believe that the non-common optical path configuration caused the drift, which will be discussed in the next section. On the contrary, the curve of the strain gauge is blurred. These small-range test results indicate that our system has the capability of measuring nanometer displacements.

 

Fig. 8 Measurement results for the step-wise motion with steps of 10 and 5 nm.

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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.
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], the minimum measurable phase can be written as:

dϕ=(ϕI1dI1)2+(ϕI2dI2)2=1I12+I22(I2dI1)2+(I1dI2)2.
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.
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.
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. 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.

 

Fig. 9 (a) Interference signals and (b) phase noises, including high- and low-frequency noises.

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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,
and
I2err=V2errsin(ϕ+ε)+S2err,
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.
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.
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. The maximum nonlinearity phase error is about 1.5° which corresponds to 2 nm of displacement error.

 

Fig. 10 Periodic nonlinearity error.

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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.
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.

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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]  

References

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  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]

2013 (2)

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]

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]

2012 (3)

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]

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]

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]

2011 (2)

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]

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]

2010 (1)

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).

2009 (3)

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]

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]

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]

2008 (3)

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).

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]

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]

2007 (1)

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]

2005 (1)

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

2004 (2)

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

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]

1999 (1)

1998 (1)

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]

1995 (1)

1994 (1)

1992 (1)

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

1988 (1)

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

1985 (1)

Arguel, P.

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]

Bojhkov, B.

Bonnefont, S.

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]

Chang, C. C.

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

Chang, R. S.

Chang, W. Y.

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]

Chen, H. Y.

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]

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]

Chen, J. H.

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]

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]

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]

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]

Chen, K.

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]

Chen, K. H.

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]

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]

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]

Chen, W. W.

Chen, Y. C.

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]

Chen, Y. Z.

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]

Cheng, C. H.

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]

Chiu, H. S.

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]

Chiu, M. H.

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]

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]

Demarest, F. C.

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]

Dian, S.

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]

Estler, W. T.

Fan, K. C.

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).

Fourment, S.

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]

Gao, W.

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]

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]

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).

Hosono, K.

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]

Hsu, C. C.

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]

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]

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]

Huang, S. H.

Ishii, Y.

Ito, S.

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]

Jay, J.

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]

Jourlin, Y.

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]

Kao, C. F.

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).

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

Kao, F. H.

Kim, W. J.

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]

Kimura, A.

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]

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).

Lai, C. W.

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]

Lee, J. Y.

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]

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]

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]

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]

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]

Li, X.

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]

Lin, J. Y.

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]

Lin, K. Y.

Lozes, F.

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]

Lu, M. H.

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

Lu, M. P.

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]

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]

Lu, S. H.

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]

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).

Moffat, R. J.

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

Muto, H.

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]

Nevièvre, M.

Noullet, J. L.

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]

Onodera, R.

Parriaux, O.

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]

Popov, E.

Sarrabayrouse, G.

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]

Shen, H. M.

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).

Shi, L.

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]

Shih, B. Y.

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]

Shimizu, Y.

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]

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]

Shyu, L. H.

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]

Teimel, A.

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

Tonchev, S.

Tsonev, L.

Wang, S. F.

Weng, H. F.

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]

Wu, C. C.

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]

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]

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]

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]

Wu, C. M.

Wu, T. H.

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]

Yang, J. S.

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]

Yang, T. H.

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]

Zeng, L.

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]

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).

Appl. Opt. (6)

Exp. Therm. Fluid Sci. (1)

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

J. Appl. Phys. (1)

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).

Meas. (1)

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]

Meas. Sci. Technol. (1)

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]

Opt. Commun. (4)

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]

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]

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]

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]

Opt. Eng. (2)

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]

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

Opt. Lasers Eng. (1)

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]

Opt. Lett. (1)

Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder (1)

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).

Precis. Eng. (3)

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]

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]

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

Sens. Actuators A Phys. (3)

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]

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]

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]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the wavelength phase-shifting grating interferometer. The diffraction beams can be reflected by (a) the mirrors or (b) the corner cube retro-reflectors. These reflected beams are diffracted by the grating G again, and interfere with each other. FG: Function Generator, LD: Laser Diode, BS: Beam Splitter, G: Grating, M: Mirror, C: Corner Cube Retro-reflectors, PZT: Piezoelectric actuators, PD: Photodetector, PC: Personal Computer.

Fig. 2
Fig. 2

Simulated interference signal intensity which is a function of the injection current.

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.

Fig. 4
Fig. 4

Lissajous patterns of (a) the original (I1 and I2), and (b) modified (I'1 and I'2) interference signals.

Fig. 5
Fig. 5

Measurement results for a long displacement of ~1 mm. Red curves: measured displacements, blue curves: measured by the encoder. Curves are offset by a few seconds for convenience of observation.

Fig. 6
Fig. 6

Measurement results for forward and backward displacement with amplitudes of about 50, 20, 10 and 5 μm.

Fig. 7
Fig. 7

Measurement results for the step-wise motion with step of 50 and 25 nm.

Fig. 8
Fig. 8

Measurement results for the step-wise motion with steps of 10 and 5 nm.

Fig. 9
Fig. 9

(a) Interference signals and (b) phase noises, including high- and low-frequency noises.

Fig. 10
Fig. 10

Periodic nonlinearity error.

Equations (27)

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E ±1 exp( i 2π λ l ±1 ±i ϕ g ).
E ' ±1 exp( i 2π λ 2 l ±1 ±i2 ϕ g ).
I | E ' +1 +E ' 1 | 2 =1+cos( 2πΔl /λ +ϕ ),
ϕ=4 ϕ g =8πΔx/Λ,
Δx=( Λ/8π )ϕ.
I( t )S( t )[ 1+Vcos( 2πΔl / λ( t ) +ϕ ) ],
I 1 S 1 [ 1+Vcos( 2πΔl / λ 1 +ϕ ) ], 0<t<T/2 ,
I 2 S 2 [ 1+Vcos( 2πΔl / λ 2 +ϕ ) ], T/2 <t<T,
I 2 S 2 [ 1+Vcos( 2π λ 1 Δl 2π λ 1 2 ΔλΔl+ϕ ) ]= S 2 [ 1+Vsin( 2π λ 1 Δl+ϕ ) ],
I( i )=( S 0 + m s i )[ 1+Vcos( 2πΔl / ( λ 1 + m λ i ) +ϕ ) ],
I 1min = S 1 ( 1V ),
I 1max = S 1 ( 1+V ).
S 1 =( I 1min + I 1max )/2.
S 2 =( I 2min + I 2max )/2.
I 1 =( I 1 S 1 )/ S 1 =Vcos( 2πΔl / λ 1 +ϕ ),
I 2 =( I 2 S 2 )/ S 2 =Vsin( 2πΔl / λ 1 +ϕ ).
ϕ= tan 1 ( I 2 / I 1 )2πΔl/ λ 1 .
e= I 1 2 + I 2 2 .
s= dϕ dΔx = 8π Λ .
dϕ= ( ϕ I 1 d I 1 ) 2 + ( ϕ I 2 d I 2 ) 2 = 1 I 1 2 + I 2 2 ( I 2 d I 1 ) 2 + ( I 1 d I 2 ) 2 .
dϕ= [ ( I 2 S 2 )d I 1 ] 2 + [ ( I 1 S 1 )d I 2 ] 2 S 2 ( I 1 S 1 ) 2 / S 1 + S 1 ( I 2 S 2 ) 2 / S 2 .
dϕ= dI SV .
I 1err = V 1err cos( ϕ )+ S 1err ,
I 2err = V 2err sin( ϕ+ε )+ S 2err ,
δϕ= ϕ err ϕ= tan 1 [ V 2err sin( ϕ+ε )+ S 2err V 1err cos( ϕ )+ S 1err ]ϕ.
ε= 8π Λ u 2f .
dϕ dt = 8π Λ dΔx dt = 8π Λ uπ f ODR .

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