Abstract

In fiber-optic interferometers with laser frequency modulation, carrier phase delay and accompanied optical intensity modulation (AOIM) in phase-generated-carrier (PGC) demodulation inevitably produce nonlinear errors that can seriously hamper displacement measurement accuracy. As for the existing improved PGC scheme, they are only capable to compensate for one of these effects. As the only method that is effective in eliminating the two effects simultaneously, typical ellipse fitting methods require target movements λlaser/4, and fail when the PGC carrier phase delay is proximate to certain values (e.g., nπ +π/4, nπ +π/2). Herein, a modified nonlinear-error correction method for errors due to PGC carrier phase delay and AOIM is proposed. Active laser-wavelength scanning by constant variation of the laser drive temperature is used to replace the target movement. A fiber-optic Michelson interferometer is constructed and experiments are performed to verify the feasibility of the proposed method. The experimental results show that after correction, the nonlinear error is reduced to less than 1nm, and nanoscale displacement measurement is achieved.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

2017 (1)

2015 (2)

S. Zhang, A. Zhang, and H. Pan, “Eliminating light intensity disturbance with reference compensation in interferometers,” IEEE Photonics Technol. Lett. 27(17), 1888–1891 (2015).
[Crossref]

W. Xia, Q. Liu, H. Hao, D. Guo, M. Wang, and X. Chen, “Sinusoidal phase-modulating self-mixing interferometer with nanometer resolution and improved measurement velocity range,” Appl. Opt. 54(26), 7820–7827 (2015).
[Crossref]

2013 (1)

2012 (1)

P. G. Jia and D. H. Wang, “Self-calibrated non-contact fibre-optic Fabry–Perot interferometric vibration displacement sensor system using laser emission frequency modulated phase generated carrier demodulation scheme,” Meas. Sci. Technol. 23(11), 115–201 (2012).
[Crossref]

2011 (1)

T. Požar, P. Gregorčič, and J. Možina, ““A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer,” Appl. Phys. B 105(3), 575–582 (2011).
[Crossref]

2010 (1)

2009 (2)

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

2005 (1)

2004 (1)

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

2003 (1)

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

1999 (1)

1996 (1)

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

1994 (2)

T. R. Christian, P. A. Frank, and B. H. Houston, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” Proc. SPIE 2191, 324–336 (1994).
[Crossref]

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3-4), 246–248 (1994).
[Crossref]

1982 (1)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Tech. 30(10), 1635–1641 (1982).
[Crossref]

1981 (1)

Bu, P.

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

Chen, B.

Chen, X.

Chen, Z.

Christian, T. R.

T. R. Christian, P. A. Frank, and B. H. Houston, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” Proc. SPIE 2191, 324–336 (1994).
[Crossref]

Chu, X. H.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Dai, G.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

Dandridge, A.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Tech. 30(10), 1635–1641 (1982).
[Crossref]

Danzebrink, H.-U.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

Duan, F.

Frank, P. A.

T. R. Christian, P. A. Frank, and B. H. Houston, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” Proc. SPIE 2191, 324–336 (1994).
[Crossref]

Giallorenzi, T. G.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Tech. 30(10), 1635–1641 (1982).
[Crossref]

Gregorcic, P.

T. Požar, P. Gregorčič, and J. Možina, ““A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer,” Appl. Phys. B 105(3), 575–582 (2011).
[Crossref]

Guo, D.

Hao, H.

Hasche, K.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

He, J.

Heydemann, P. L. M.

Houston, B. H.

T. R. Christian, P. A. Frank, and B. H. Houston, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” Proc. SPIE 2191, 324–336 (1994).
[Crossref]

Huang, B.

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

Jackson, D. A.

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3-4), 246–248 (1994).
[Crossref]

Jia, P. G.

P. G. Jia and D. H. Wang, “Self-calibrated non-contact fibre-optic Fabry–Perot interferometric vibration displacement sensor system using laser emission frequency modulated phase generated carrier demodulation scheme,” Meas. Sci. Technol. 23(11), 115–201 (2012).
[Crossref]

Kireenkov, A. Y.

A. N. Nikitenko, M. Y. Plotnikov, A. V. Volkov, M. V. Mekhrengin, and A. Y. Kireenkov, “PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors,” IEEE Sensors J. 18(5), 1985–1992 (2018).
[Crossref]

Lai, S. R.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Li, F.

Li, Z.

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

Liao, Y.

Liao, Y. B.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Liu, Q.

Liu, Y.

J. He, L. Wang, F. Li, and Y. Liu, “An ameliorated phase generated carrier demodulation algorithm with low harmonic distortion and high stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).
[Crossref]

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

Lou, Y.

Maruyama, T.

Matsuda, M.

McGarrity, C.

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3-4), 246–248 (1994).
[Crossref]

Mekhrengin, M. V.

A. N. Nikitenko, M. Y. Plotnikov, A. V. Volkov, M. V. Mekhrengin, and A. Y. Kireenkov, “PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors,” IEEE Sensors J. 18(5), 1985–1992 (2018).
[Crossref]

Možina, J.

T. Požar, P. Gregorčič, and J. Možina, ““A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer,” Appl. Phys. B 105(3), 575–582 (2011).
[Crossref]

Nikitenko, A. N.

A. N. Nikitenko, M. Y. Plotnikov, A. V. Volkov, M. V. Mekhrengin, and A. Y. Kireenkov, “PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors,” IEEE Sensors J. 18(5), 1985–1992 (2018).
[Crossref]

Pan, H.

S. Zhang, A. Zhang, and H. Pan, “Eliminating light intensity disturbance with reference compensation in interferometers,” IEEE Photonics Technol. Lett. 27(17), 1888–1891 (2015).
[Crossref]

Plotnikov, M. Y.

A. N. Nikitenko, M. Y. Plotnikov, A. V. Volkov, M. V. Mekhrengin, and A. Y. Kireenkov, “PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors,” IEEE Sensors J. 18(5), 1985–1992 (2018).
[Crossref]

Pohlenz, F.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

Požar, T.

T. Požar, P. Gregorčič, and J. Možina, ““A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer,” Appl. Phys. B 105(3), 575–582 (2011).
[Crossref]

Ruffin, P. B.

S. Yin, P. B. Ruffin, and F. T. S. Yu, Fiber Optic Sensors, 2nd ed. CRC University, Boca Raton, FL, USA, 2008.

Sasaki, O.

Su, C.-S.

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Suzuki, T.

Tan, S.

Tian, C. D.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Tveten, A. B.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Tech. 30(10), 1635–1641 (1982).
[Crossref]

Volkov, A. V.

A. N. Nikitenko, M. Y. Plotnikov, A. V. Volkov, M. V. Mekhrengin, and A. Y. Kireenkov, “PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors,” IEEE Sensors J. 18(5), 1985–1992 (2018).
[Crossref]

Wang, D. H.

P. G. Jia and D. H. Wang, “Self-calibrated non-contact fibre-optic Fabry–Perot interferometric vibration displacement sensor system using laser emission frequency modulated phase generated carrier demodulation scheme,” Meas. Sci. Technol. 23(11), 115–201 (2012).
[Crossref]

Wang, K.

Wang, L.

Wang, L. W.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Wang, M.

Wang, X.

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

Wilkening, G.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

Wu, C.-M.

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Xia, W.

Xie, J.

Xie, S.

Xu, Z.

Yan, L.

Yin, S.

S. Yin, P. B. Ruffin, and F. T. S. Yu, Fiber Optic Sensors, 2nd ed. CRC University, Boca Raton, FL, USA, 2008.

Yu, D.

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

Yu, F. T. S.

S. Yin, P. B. Ruffin, and F. T. S. Yu, Fiber Optic Sensors, 2nd ed. CRC University, Boca Raton, FL, USA, 2008.

Zhang, A.

S. Zhang, A. Zhang, and H. Pan, “Eliminating light intensity disturbance with reference compensation in interferometers,” IEEE Photonics Technol. Lett. 27(17), 1888–1891 (2015).
[Crossref]

Zhang, C.

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

Zhang, H. Y.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Zhang, M.

K. Wang, M. Zhang, F. Duan, S. Xie, and Y. Liao, “Measurement of the phase shift between intensity and frequency modulations within DFB-LD and its influences on PGC demodulation in a fiber-optic sensor system,” Appl. Opt. 52(29), 7194–7199 (2013).
[Crossref]

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Zhang, S.

Zhang, Y.

Zheng, D.

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

Appl. Opt. (4)

Appl. Phys. B (1)

T. Požar, P. Gregorčič, and J. Možina, ““A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer,” Appl. Phys. B 105(3), 575–582 (2011).
[Crossref]

IEEE Photonics Technol. Lett. (1)

S. Zhang, A. Zhang, and H. Pan, “Eliminating light intensity disturbance with reference compensation in interferometers,” IEEE Photonics Technol. Lett. 27(17), 1888–1891 (2015).
[Crossref]

IEEE Sensors J. (1)

A. N. Nikitenko, M. Y. Plotnikov, A. V. Volkov, M. V. Mekhrengin, and A. Y. Kireenkov, “PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors,” IEEE Sensors J. 18(5), 1985–1992 (2018).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Tech. 30(10), 1635–1641 (1982).
[Crossref]

J. Lightwave Technol. (1)

Meas. Sci. Technol. (3)

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[Crossref]

P. G. Jia and D. H. Wang, “Self-calibrated non-contact fibre-optic Fabry–Perot interferometric vibration displacement sensor system using laser emission frequency modulated phase generated carrier demodulation scheme,” Meas. Sci. Technol. 23(11), 115–201 (2012).
[Crossref]

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Opt. Commun. (1)

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3-4), 246–248 (1994).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

X. Wang, X. Wang, Y. Liu, C. Zhang, and D. Yu, “A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source,” Opt. Laser Technol. 35(3), 219–222 (2003).
[Crossref]

Optik (1)

Z. Li, X. Wang, P. Bu, B. Huang, and D. Zheng, “Sinusoidal phase-modulating laser diode interferometer insensitive to the intensity modulation of the light source,” Optik 120(16), 799–803 (2009).
[Crossref]

Proc. SPIE (2)

T. R. Christian, P. A. Frank, and B. H. Houston, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” Proc. SPIE 2191, 324–336 (1994).
[Crossref]

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[Crossref]

Other (2)

S. Yin, P. B. Ruffin, and F. T. S. Yu, Fiber Optic Sensors, 2nd ed. CRC University, Boca Raton, FL, USA, 2008.

E. Udd and W. B. Spillman, eds., Fiber Optic Sensors: An Introduction for Engineers and Scientists. Wiley, Hoboken, NJ, USA, 2011.

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Figures (9)

Fig. 1.
Fig. 1. Schematic of fiber-optic Michelson interferometer with laser frequency modulation: FC: fiber optic circulator; SMF: single mode fiber; BS: beam splitter; M1, M2: mirrors 1 and 2, respectively; APD: avalanche photodetector; DDS1, DDS2: direct digital synthesizer 1 and 2, respectively; LPF: low-pass filter; DFB: distributed feedback laser; GRIN: gradient-index lens; ADC: analog-to-digital converter; DAC: digital-to- analog converter.
Fig. 2.
Fig. 2. Effects of carrier phase delay and AOIM in PGC demodulation: (a)–(d) two output signals (red and green traces) from a LPF and Lissajous figure (blue traces) of S1(t) and S2(t) for the carrier phase delays of 0, π/4, π/2, and 3π/4, respectively..
Fig. 3.
Fig. 3. Lissajous figures (different laser drive temperature and current): (a) driving current is constant and driving temperature is variable, and (b) driving current is variable and driving temperature is constant
Fig. 4.
Fig. 4. Schematic of nonlinearity correction in fiber-optic Michelson interferometers with laser frequency modulation. M1, M2: mirrors 1 and 2, respectively; APD: avalanche photodetector; BS: beam splitter; SMF: single mode fiber; FC: fiber optic circulator; DDS1, DDS2: direct digital synthesizer 1 and 2, respectively; ADC: analog-to-digital converter; LPF: low-pass filter; DAC: digital-to- analog converter; DFB: distributed feedback laser; GRIN: gradient-index lens; EN: enable end of a nonlinearity correction unit; INT: integer processing unit; Pick: optimal value picking.
Fig. 5.
Fig. 5. Lissajous trajectories with target displacement of (a) < λ/4 and (b) ≥λ/4. (c) Schematic of gain/offset correction method based on peak value extraction. PVD: peak value detector.
Fig. 6.
Fig. 6. Lissajous figures with characteristic parameters obtained during (a) temperature scanning and (b) target motion. A, B, S1c, S2c, α: characteristic parameters of ellipse.
Fig. 7.
Fig. 7. Nonlinear error precompensation results: (a, c) approximate calculation of carrier phase delays, where S3(t) (red traces) and S4(t) (pink traces) are correspond to the left ordinate and calculated carrier phase delays are φm(t) (blue traces) correspond to the right ordinate, and (b, d) comparison of Lissajous figures of S1(t) and S2(t) before and after precompensation when the phase delay was set to π/4 and π/2, respectively.
Fig. 8.
Fig. 8. Experimental results of nonlinear error fine correction for carrier phase delays of (a) π/4 and (b) π/2. Note that demodulation phase values (blue traces) are correspond to the left ordinate and nonlinear errors (red traces) are correspond to the right ordinate, nonlinear error fine correction is enable at 3 ms.
Fig. 9.
Fig. 9. Experimental results of displacement measurements at the nanoscale. (a) Measured results for 5 nm steps conducted 600 times (total range = 3 µm). Note that the red line is shifted by 0.5 µm to allow plot visibility, demodulation displacement (black traces) and displacement of the stage (red traces) are correspond to the left ordinate, deviation (blue traces) are correspond to the right ordinate. (b) FFT analysis of displacement deviation.

Tables (1)

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Table 1. Comparison of elliptic characteristic parameters obtained based on elliptic fitting.

Equations (32)

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S ( t ) = k I 0 [ 1 + v cos ( C cos ω 0 t + φ ( t ) ) ] .
S ( t ) = k I 0 + k I 0 v { cos [ C cos ( ω 0 t ) ] cos φ ( t )  -  sin [ C cos ( ω 0 t ) ] sin φ ( t ) } .
S ( t ) = k I 0 + k I 0 v { [ J 0 ( C ) + 2 k = 1 ( 1 ) k J 2 k ( C ) cos 2 k ω 0 t ] cos φ ( t ) 2 [ k = 0 ( 1 ) k J 2 k + 1 ( C ) cos ( 2 k + 1 ) ω 0 t ] sin φ ( t ) } .
S ( t ) ω 0 = 2 k I 0 v J 1 ( C ) cos ( ω 0 t ) sin φ ( t ) ,
S ( t ) 2 ω 0 = 2 k I 0 v J 2 ( C ) cos ( 2 ω 0 t ) cos φ ( t ) .
U m 1 = k I 0 v J 1 ( C ) sin φ ( t ) ,
U m 2 = k I 0 v J 2 ( C ) cos φ ( t ) ,
H ( t ) = arctan U m 1 U m 2 = arctan J 1 ( C ) sin φ ( t ) J 2 ( C ) cos φ ( t ) .
φ ( t ) = arctan U m 1 U m 2 = arctan sin φ ( t ) cos φ ( t )  =  arctan ( tan φ ( t ) ) .
S ( t ) = k I 0 [ 1 + m cos ( ω 0 ( t τ ) + φ m ) ] [ 1 + v cos ( C cos ω 0 ( t τ ) + φ ( t ) ) ] ,
S 1 ( t ) = k I 0 v P 1 [ sin ( φ ( t ) θ 1 ) ] + m k I 0 P 3 ,
S 2 ( t ) = k I 0 v P 2 [ cos ( φ ( t ) θ 2 ) ] ,
P 1 = { m / m 2 2 [ J 0 ( C ) cos ( φ c φ m ) J 2 ( C ) cos ( φ c + φ m ) ] } 2 + [ J 1 ( C ) cos φ c ] 2 ,
P 2 = { m / m 2 2 [ J 3 ( C ) cos ( 2 φ c  +  φ m ) J 1 ( C ) cos ( 2 φ c φ m ) ] } 2 + [ J 2 ( C ) cos ( 2 φ c ) ] 2 ,
P 3 = 1 / 1 2 cos ( φ 2 cos ( φ c φ m ) ,
tan θ 1 = m [ J 0 ( C ) cos ( φ c φ m ) J 2 ( C ) cos ( φ c + φ m ) ] / m [ J 0 ( C ) cos ( φ c φ m ) J 2 ( C ) cos ( φ c + φ m ) ] [ J 1 ( C ) cos φ c [ J 1 ( C ) cos φ c ] ,
tan θ 2 = m [ J 3 ( C ) cos ( 2 φ c  +  φ m ) J 1 ( C ) cos ( 2 φ c φ m ) ] / m [ J 3 ( C ) cos ( 2 φ c  +  φ m ) J 1 ( C ) cos ( 2 φ c φ m ) ] [ J 2 ( C ) cos 2 φ c [ J 2 ( C ) cos 2 φ c ] .
φ e r r o r ( t )  +  φ ( t ) = arctan ( S 1 ( t ) / S 1 ( t ) S 2 ( t ) S 2 ( t ) ) .
φ ( t ) = φ 0 + φ s ( t ) = 4 π n L λ + 4 π n x ( t ) λ .
φ ( λ , t ) = 4 π n ( L + x ( t ) ) λ .
Δ φ = φ ( λ 1 , t 1 ) φ ( λ 2 , t 2 ) = 4 π n ( L + x ( t 1 ) ) λ 1 4 π n ( L + x ( t 2 ) ) λ 2 ,
Δ φ  =  ( λ 2 λ 1 ) λ 2 λ 1 4 π n L 2 π ,
S 3 ( t ) = k I 0 v J 1 cos φ c sin φ ( t ) ,
S 4 ( t ) = k I 0 v J 1 sin φ c sin φ ( t ) ,
φ c ( t ) = atan 2 ( S 4 ( t ) , S 3 ( t ) ) ,
φ c ( t ) = { I N T ( π φ c ( t ) π / 180 ) φ c ( t ) 0 I N T ( φ c ( t ) π / 180 ) φ c ( t ) < 0 .
S 1 ( t ) = S 1 ( t ) ( S 1 max + S 1 min ) / S 1 ( t ) ( S 1 max + S 1 min ) 2 2 ( S 1 max S 1 min ) / ( S 1 max S 1 min ) 2 2  =  sin ( φ ( t ) θ 1 ) ,
S 2 ( t ) = S 2 ( t ) ( S 2 max + S 2 min ) / S 2 ( t ) ( S 2 max + S 2 min ) 2 2 ( S 2 max S 2 min ) / ( S 2 max S 2 min ) 2 2  =  cos ( φ ( t ) θ 2 ) .
S 1 ( t ) = S 1 ( t ) S 2 (t) = 2 cos ( π 4 + θ 1 θ 2 2 ) sin [ φ ( t ) ( π 4 + θ 1 + θ 2 2 ) ] ,
S 2 ( t ) = S 1 ( t )  +  S 2 (t) = 2 sin ( π 4 + θ 1 θ 2 2 ) cos [ φ ( t ) ( π 4 + θ 1 + θ 2 2 ) ] ,
S 1 ( t ) = S 1 ( t ) ( S 1 max S 1 min ) / ( S 1 max S 1 min ) 2 2  =  sin [ φ ( t ) ( π 4 + θ 1 + θ 2 2 ) ] ,
S 2 ( t ) = S 2 ( t ) ( S 2 max S 2 min ) / ( S 2 max S 2 min ) 2 2  =  cos [ φ ( t ) ( π 4 + θ 1 + θ 2 2 ) ] .

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