## Displacement measurement using a wavelength-phase-shifting grating interferometer |

Optics Express, Vol. 21, Issue 21, pp. 25553-25564 (2013)

http://dx.doi.org/10.1364/OE.21.025553

Acrobat PDF (1939 KB)

### 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

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]

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]

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]

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

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]

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

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]

*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. 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]

## 2. Principle

### 2.1 Double-diffraction interference system

*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

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]

*x*axis by an amount Δ

*x*, the optical phase in the + 1st- and −1st-order beams increases and decreases, respectively, by

*ϕ*= 2

_{g}*π*Δ

*x*/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:Here 2

*π*/

*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 M

_{1}and M

_{2}, respectively. Then, these two diffraction beams are reflected from M

_{1}and M

_{2}, and diffracted again by the grating G. These two double-diffracted beams can be expressed as:

*l*= 2(

*l*

_{+1}

*− l*

_{−1}) is the optical path difference of the two double-diffracted beams.

*ϕ*= 2

*ϕ*− (−2

_{g}*ϕ*) = 4

_{g}*ϕ*is the phase variation of the interference signal, which is 4 times the optical phase variation of the diffracted beams. The optical path difference Δ

_{g}*l*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 M

_{1}and M

_{2}can be replaced by the corner cube retro-reflectors C

_{1}and C

_{2}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 Δ

*x*is given as:orIt is obvious that the grating displacement Δ

*x*can be determined by measuring the phase variation

*ϕ*of the interference signal.

### 2.2 Wavelength modulation technique and quadrature method for phase detection

*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: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 (

*λ*

_{1}and

*λ*

_{2}) sequentially in one period. The sequential interference signal can be expressed as:andwhere

*S*

_{1}and

*S*

_{2}are the main intensities of the interference signals. Here we can select a suitable

*λ*

_{2}=

*λ*

_{1}+ Δ

*λ*to make a

*π*/2 phase difference between

*I*

_{2}and

*I*

_{1}, that is:where Δ

*λ*= (

*λ*

_{2}−

*λ*

_{1}) <<

*λ*

_{1}, and 4Δ

*λ*Δ

*l*/

*λ*

_{1}

^{2}= 1. The interference signals

*I*

_{1}and

*I*

_{2}(Eqs. (6a) and (7)) are quadrature. By adjusting the DC (

*S*

_{1}and

*S*

_{2}) and AC (

*S*

_{1}

*V*and

*S*

_{2}

*V*) terms, these two signals can be used to solve the phase difference

*ϕ*.

### 2.3 Selection of the wavelengths λ_{2} and λ_{1}

*λ*

_{2}and

*λ*

_{1}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:where

*m*and

_{s}*m*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

_{λ}*I*

_{a}and maximum

*I*

_{b}at

*i*

_{a}and

*i*

_{b}, respectively. Of course, the phase difference between

*I*

_{a}and

*I*

_{b}is ~

*π*. We can estimate that the phase difference

*π*/2 will occur at

*i*

_{ab}= (

*i*

_{a}+

*i*

_{b})/2, and we assume that the wavelengths

*λ*

_{1}and

*λ*

_{2}correspond to the injection currents

*i*

_{a}and

*i*

_{ab}, respectively. Then we set the injection current at

*i*

_{a}for wavelength

*λ*

_{1}and give the grating a sufficient displacement. As shown in Fig. 3, the intensity of the interference signal oscillates between the minimum

*I*

_{1min}and maximum

*I*

_{1max}(see the upper curve in Fig. 3) because the phase

*ϕ*increases (or decreases). The minimum

*I*

_{1min}and maximum

*I*

_{1max}in Eq. (6a) can be expressed as:andThe main intensity (or DC term) of the interference signal

*I*

_{1}is:Similarly, the main intensity

*S*

_{2}of the interference signal

*I*

_{2}(see the lower curve in Fig. 3) can be given as:In order to determine the phase variation

*ϕ*, the sequential interference signals in Eqs. (6a) and (7) can be processed as:andThe relationship between the phase variation

*ϕ*and the modified interference signals is:

*I'*

_{1}and

*I'*

_{2}. The last term can be ignored if the optical path difference Δ

*l*is constant. Substituting the measured phase variation

*ϕ*into Eq. (4b), the grating displacement Δ

*x*can be obtained. The curves

*a*and

*b*in Fig. 4 show the Lissajous patterns of the original (

*I*

_{1}and

*I*

_{2}) 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: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

*S*

_{1},

*S*

_{2},

*i*

_{a}, and

*i*

_{ab}until the error signal

*e*is a constant, and a much purer Lissajous circle can be obtained.

## 3. Performance test

### 3.1 Experimental setup

### 3.2 Millimeter-scale displacement testing

### 3.3 Micrometer-scale displacement testing

### 3.4 Nanometer-scale displacement testing

*-*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.

## 4. Discussion

### 4.1 Measurement sensitivity

### 4.2 Measurement resolution and stability

*d*Δ

*x*=

*dϕ*/

*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]

*I'*

_{1}and

*I'*

_{2}from Eqs. (11) and (12) into Eq. (16),

*dϕ*can be written as:Here

*dI*

_{1}and

*dI*

_{2}are the minimum detectable intensity of the detector, and they have the same magnitude

*dI*=

*dI*

_{1}=

*dI*

_{2}. In our experiments,

*S*

_{1}is nearly equal to

*S*

_{2}. From Eqs. (11) and (12), Eq. (17) can be simplified to:Here we set the main intensity

*S = S*

_{1}=

*S*

_{2}.

*I*

_{1}and

*I*

_{2}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 Δ

*l*in the last term of Eqs. (6a) and (6b) is the drift source of

*I*

_{1}and

*I*

_{2}. The low frequency noises can be suppressed in the good experimental environment. Not only the low frequency noises, these curves of

*I*

_{1},

*I*

_{2}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

*I*

_{1},

*I*

_{2}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

*dϕ*= 0.028 rad (~1.6°) is obtained. From Eq. (15), the minimum measurable displacement or measurement resolution of our system is

*d*Δ

*x*=

*dϕ*/

*s*≈2 nm.

### 4.3 Periodic nonlinearity error analysis

*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:andwhere

*V*

_{1}

*and*

_{err}*V*

_{2}

*are the unequal AC terms,*

_{err}*S*

_{1}

*and*

_{err}*S*

_{2}

*are the residual DC terms or biases, and*

_{err}*ε*is the phase-shifting error. Here the phase 2

*π*Δ

*l*/

*λ*

_{1}is ignored. It is obvious that the Lissajous patterns of

*I'*

_{1}

*and*

_{err}*I'*

_{2}

*will ellipses, and the calculated phase*

_{err}*ϕ*will suffer from the nonlinear error

_{err}*δϕ*:According to the experimental estimation in our measurement system, the worst case for the ratio of

*V*

_{2}

*to*

_{err}*V*

_{1}

*is about 1.005, and biases*

_{err}*S*

_{1}

*and*

_{err}*S*

_{2}

*both are about 0.005. Actually, the optical phase variation which results from the moving grating will bring about the phase-shifting error*

_{err}*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: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.

### 4.4 Limitation of measurement speed

*dϕ*/

*dt*of the optical phase and the output data rate

*f*

_{ODR}of the PC-based program can be written aswhere

*u*stands for the speed of the moving grating. The output data rate

*f*

_{ODR}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

## Acknowledgments

## References and links

1. | C. M. Wu, “Heterodyne interferometric system with sub-nanometer accuracy for measurement of straightness,” Appl. Opt. |

2. | F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. |

3. | W. T. Estler, “High-accuracy displacement interferometry in air,” Appl. Opt. |

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. |

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. |

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. |

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. |

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. |

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. |

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. |

11. | A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. |

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. |

13. | C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng. |

14. | J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun. |

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. |

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. |

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. |

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. |

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. |

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. |

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. |

22. | R. Onodera and Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt. |

23. | R. Onodera and Y. Ishii, “Two-wavelength laser-diode heterodyne interferometry with one phasemeter,” Opt. Lett. |

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. |

25. | R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci. |

**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

- C. M. Wu, “Heterodyne interferometric system with sub-nanometer accuracy for measurement of straightness,” Appl. Opt.43(19), 3812–3816 (2004). [CrossRef] [PubMed]
- 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]
- W. T. Estler, “High-accuracy displacement interferometry in air,” Appl. Opt.24(6), 808–815 (1985). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng.14(3), 147–154 (1992). [CrossRef]
- 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]
- C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng.44, 023063 (2005).
- 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. 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. 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]
- 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).
- 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]
- 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]
- 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]
- R. Onodera and Y. Ishii, “Two-wavelength laser-diode heterodyne interferometry with one phasemeter,” Opt. Lett.20(24), 2502–2504 (1995). [CrossRef] [PubMed]
- 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]
- R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci.1(1), 3–17 (1988). [CrossRef]

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