Abstract

A self-mixing birefringent dual-frequency laser Doppler velocimeter (SBD-LDV) for high-resolution velocity measurements is presented in this paper. The velocity information of the object can be accurately extracted from the self-mixing Doppler frequency shift of the birefringent light-carried microwave signal. We generate a virtual stable light-carried microwave by using a birefringent dual-frequency He-Ne laser which further simplifies the structure of the light source. Moreover, the optical configuration based on the laser self-mixing interference brings benefits of compact optical setup, self-alignment, and direction discriminability. Experimentally, we extracted the Doppler beat frequency signal by the low-frequency (millihertz) phase lock-in amplifier, measured the beat frequency precisely in time-domain with a low sampling rate and calculated the magnitude of velocity. Compared with the previous self-mixing LDV, the average velocity resolution of SBD-LDV is improved to 0.030 mm/s for a target with longitudinal velocity, benefiting from the high stability of light-carried microwave. It is of great meaning and necessity because it helps to provide an available velocimeter with high stability and an extremely compact configuration, making a potential contribution to the velocimetry in practical engineering application.

© 2017 Optical Society of America

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References

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  1. L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
    [Crossref]
  2. S. K. Ozdemir, I. Ohno, and S. Shinohara, “A comparative study for the assessment on blood flow measurement using self-mixing laser speckle interferometer,” IEEE Trans. Instrum. Meas. 57(2), 355–363 (2008).
    [Crossref]
  3. L. Rovati, S. Cattini, and N. Palanisamy, “Measurement of the fluid-velocity profile using a self-mixing super luminescent diode,” Meas. Sci. Technol. 22(2), 025402 (2011).
    [Crossref]
  4. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
    [Crossref]
  5. W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
    [Crossref]
  6. R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5-µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
    [Crossref]
  7. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
    [Crossref]
  8. C. H. Cheng, C. W. Lee, T. W. Lin, and F. Y. Lin, “Dual-frequency laser Doppler velocimeter for speckle noise reduction and coherence enhancement,” Opt. Express 20(18), 20255–20265 (2012).
    [Crossref] [PubMed]
  9. C. H. Cheng, L. C. Lin, and F. Y. Lin, “Self-mixing dual-frequency laser Doppler velocimeter,” Opt. Express 22(3), 3600–3610 (2014).
    [Crossref] [PubMed]
  10. S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183(1–4), 195–205 (2000).
    [Crossref]
  11. Y. H. Liao and F. Y. Lin, “Dynamical characteristics and their applications of semiconductor lasers subject to both optical injection and optical feedback,” Opt. Express 21(20), 23568–23578 (2013).
    [Crossref] [PubMed]
  12. H. Zhu, J. Chen, D. Guo, W. Xia, H. Hao, and M. Wang, “Birefringent dual-frequency laser Doppler velocimeter using a low-frequency lock-in amplifier technique for high-resolution measurements,” Appl. Opt. 55(16), 4423–4429 (2016).
    [Crossref] [PubMed]
  13. L. Fei and S. Zhang, “Self-mixing interference effects of orthogonally polarized dual frequency laser,” Opt. Express 12(25), 6100–6105 (2004).
    [Crossref] [PubMed]
  14. S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68(1), 55–57 (1988).
    [Crossref]
  15. P. Zhang, Y. D. Tan, N. Liu, Y. Wu, and S. L. Zhang, “Phase difference in modulated signals of two orthogonally polarized outputs of a Nd:YAG microchip laser with anisotropic optical feedback,” Opt. Lett. 38(21), 4296–4299 (2013).
    [Crossref] [PubMed]
  16. Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
    [Crossref]
  17. G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
    [Crossref]
  18. B. Chen, L. Yan, X. Yao, T. Yang, D. Li, W. Dong, C. Li, and W. Tang, “Development of a laser synthetic wavelength interferometer for large displacement measurement with nanometer accuracy,” Opt. Express 18(3), 3000–3010 (2010).
    [Crossref] [PubMed]

2016 (1)

2014 (1)

2013 (2)

2012 (2)

C. H. Cheng, C. W. Lee, T. W. Lin, and F. Y. Lin, “Dual-frequency laser Doppler velocimeter for speckle noise reduction and coherence enhancement,” Opt. Express 20(18), 20255–20265 (2012).
[Crossref] [PubMed]

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

2011 (1)

L. Rovati, S. Cattini, and N. Palanisamy, “Measurement of the fluid-velocity profile using a self-mixing super luminescent diode,” Meas. Sci. Technol. 22(2), 025402 (2011).
[Crossref]

2010 (1)

2009 (1)

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
[Crossref]

2008 (1)

S. K. Ozdemir, I. Ohno, and S. Shinohara, “A comparative study for the assessment on blood flow measurement using self-mixing laser speckle interferometer,” IEEE Trans. Instrum. Meas. 57(2), 355–363 (2008).
[Crossref]

2004 (2)

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

L. Fei and S. Zhang, “Self-mixing interference effects of orthogonally polarized dual frequency laser,” Opt. Express 12(25), 6100–6105 (2004).
[Crossref] [PubMed]

2003 (1)

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

2002 (1)

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
[Crossref]

2000 (1)

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183(1–4), 195–205 (2000).
[Crossref]

1994 (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
[Crossref]

1988 (1)

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68(1), 55–57 (1988).
[Crossref]

1986 (1)

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5-µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[Crossref]

Bosch, T.

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
[Crossref]

Bosch, T. M.

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
[Crossref]

Boyle, W. J. O.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
[Crossref]

Cattini, S.

L. Rovati, S. Cattini, and N. Palanisamy, “Measurement of the fluid-velocity profile using a self-mixing super luminescent diode,” Meas. Sci. Technol. 22(2), 025402 (2011).
[Crossref]

Chen, B.

Chen, J.

Cheng, C. H.

Chicharo, J. F.

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
[Crossref]

Chraplyvy, A. R.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5-µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[Crossref]

Donati, S.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
[Crossref]

Dong, W.

Fei, L.

Giuliani, G.

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
[Crossref]

Grattan, K. T. V.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
[Crossref]

Guo, D.

Hao, H.

Hwang, S. K.

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183(1–4), 195–205 (2000).
[Crossref]

Lee, C. W.

Li, C.

Li, D.

Li, Y.

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Liao, Y. H.

Lin, F. Y.

Lin, L. C.

Lin, T. W.

Liu, G.

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Liu, J. M.

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183(1–4), 195–205 (2000).
[Crossref]

Liu, N.

Norgia, M.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
[Crossref]

Ohno, I.

S. K. Ozdemir, I. Ohno, and S. Shinohara, “A comparative study for the assessment on blood flow measurement using self-mixing laser speckle interferometer,” IEEE Trans. Instrum. Meas. 57(2), 355–363 (2008).
[Crossref]

Ozdemir, S. K.

S. K. Ozdemir, I. Ohno, and S. Shinohara, “A comparative study for the assessment on blood flow measurement using self-mixing laser speckle interferometer,” IEEE Trans. Instrum. Meas. 57(2), 355–363 (2008).
[Crossref]

Palanisamy, N.

L. Rovati, S. Cattini, and N. Palanisamy, “Measurement of the fluid-velocity profile using a self-mixing super luminescent diode,” Meas. Sci. Technol. 22(2), 025402 (2011).
[Crossref]

Palmer, A. W.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
[Crossref]

Plantier, G.

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

Rovati, L.

L. Rovati, S. Cattini, and N. Palanisamy, “Measurement of the fluid-velocity profile using a self-mixing super luminescent diode,” Meas. Sci. Technol. 22(2), 025402 (2011).
[Crossref]

Scalise, L.

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

Shinohara, S.

S. K. Ozdemir, I. Ohno, and S. Shinohara, “A comparative study for the assessment on blood flow measurement using self-mixing laser speckle interferometer,” IEEE Trans. Instrum. Meas. 57(2), 355–363 (2008).
[Crossref]

Tan, Y.

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

Tan, Y. D.

Tang, W.

Tkach, R. W.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5-µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[Crossref]

Wang, M.

Wang, W. M.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
[Crossref]

Wu, Y.

Xi, J.

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
[Crossref]

Xia, W.

Yan, L.

Yang, S.

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68(1), 55–57 (1988).
[Crossref]

Yang, T.

Yao, X.

Yu, Y.

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
[Crossref]

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

Zeng, Z.

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

Zhang, P.

P. Zhang, Y. D. Tan, N. Liu, Y. Wu, and S. L. Zhang, “Phase difference in modulated signals of two orthogonally polarized outputs of a Nd:YAG microchip laser with anisotropic optical feedback,” Opt. Lett. 38(21), 4296–4299 (2013).
[Crossref] [PubMed]

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

Zhang, S.

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

L. Fei and S. Zhang, “Self-mixing interference effects of orthogonally polarized dual frequency laser,” Opt. Express 12(25), 6100–6105 (2004).
[Crossref] [PubMed]

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68(1), 55–57 (1988).
[Crossref]

Zhang, S. L.

Zhu, H.

Zhu, J.

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

Z. Zeng, S. Zhang, Y. Tan, P. Zhang, and Y. Li, “Single high-order round-trip feedback effects in orthogonally polarized dual frequency laser,” Appl. Phys. B 107(2), 333–338 (2012).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009).
[Crossref]

IEEE Trans. Instrum. Meas. (2)

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004).
[Crossref]

S. K. Ozdemir, I. Ohno, and S. Shinohara, “A comparative study for the assessment on blood flow measurement using self-mixing laser speckle interferometer,” IEEE Trans. Instrum. Meas. 57(2), 355–363 (2008).
[Crossref]

J. Lightwave Technol. (2)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a singlemode diode laser for optical sensing applications,” J. Lightwave Technol. 12(9), 1577–1587 (1994).
[Crossref]

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5-µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(4), S283–S294 (2002).
[Crossref]

Meas. Sci. Technol. (1)

L. Rovati, S. Cattini, and N. Palanisamy, “Measurement of the fluid-velocity profile using a self-mixing super luminescent diode,” Meas. Sci. Technol. 22(2), 025402 (2011).
[Crossref]

Opt. Commun. (3)

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183(1–4), 195–205 (2000).
[Crossref]

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He–Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68(1), 55–57 (1988).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

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Figures (10)

Fig. 1
Fig. 1 Schematic diagram of a SBD-LDV. MT: a movable feedback mirror; ND: neutral density filter; W1, W2, W3: optical windows; PBS: polarizing beam splitter; PD1, PD2: photodiodes. The He-Ne gas discharge tube, quartz crystal and optical windows make up the birefringent dual-frequency He-Ne laser. An external plane optical window W1 with reflectivity of r1 and the concave optical window W3 of the He-Ne gas discharge tube with reflectivity of r3, form the laser resonant half-intracavity where a uniaxial quartz crystal is placed.
Fig. 2
Fig. 2 The simulation of Doppler beat frequency signal. (a) The self-mixing interference signals of two birefringently polarized modes. (b) The mixed Doppler-shifted signal. (c) The extracted difference-frequency term of the mixed signal. (d) The simulated Doppler beat frequency signal when the object is moving with the speed of 1 mm/s, 2 mm/s, 5 mm/s, 10 mm/s, 20 mm/s, 30 mm/s.
Fig. 3
Fig. 3 Configuration of velocity measurement system based on SBD-LDV.
Fig. 4
Fig. 4 The Doppler signals obtained from the SBD-LDV when the measured target is (a) approaching ( υ = + 1 mm/s) and (b) leaving ( υ = −1 mm/s) the laser. (c)The simulated power solution when υ = + 1 mm/s. (d) The simulated power solution when υ = −1 mm/s.
Fig. 5
Fig. 5 The Doppler beat frequency signal S ( t ) obtained with the SBD-LDV when the measured object is moving with the speed of (a) 1 mm/s, (b) −2 mm/s, (c) −5 mm/s, (d) 10 mm/s, (e)-30mm/s. (f) Frequency spectrum (FFT).
Fig. 6
Fig. 6 The dependence of the measured Doppler beat frequency on the actual speed of MT.
Fig. 7
Fig. 7 Comparison velocity measurement results
Fig. 8
Fig. 8 (a)The measured light-carried microwave frequency. (b) The fluctuation of the light-carried microwave frequency with feedback strength of 0.1, 0.5 and 1.
Fig. 9
Fig. 9 The frequency difference observed in a scanning Fabry-Perot interferometer. (a)- (b) With the optical feedback. (c)- (d) In the absence of optical feedback.
Fig. 10
Fig. 10 (a) Relationship between the uncertainty of the measured velocity and the sampling rate at different moving speed of the stage. (b)The uncertain of the velocity caused by the Doppler beat frequency with fixed sampling rate of 10 kHz.

Equations (11)

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Δ ν = 2 Λ h λ [ ( sin 2 θ n e 2 + cos 2 θ n o 2 ) 1 / 2 n o ] .
{ Δ G o = ln { 1 + [ r ( 1 r 1 2 ) / r 1 ] e i ω o τ E } α o cos ( 4 π ν o υ t / c + ϕ o ) Δ G e = ln { 1 + [ r ( 1 r 1 2 ) / r 1 ] e i ω e τ E } α e cos ( 4 π ν e υ t / c + ϕ e ) .
{ P o ( t ) = P o 0 [ 1 + β o cos ( 2 π f D 1 t + ϕ o ) ] P e ( t ) = P e 0 [ 1 + β e cos ( 2 π f D 2 t + ϕ e ) ] .
S ( t ) = P o 0 P e 0 β o β e cos ( 2 π f D 1 t + ϕ o ) cos ( 2 π f D 2 t + ϕ e ) = P o 0 P e 0 β o β e 2 { cos [ 2 π ( f D 1 + f D 2 ) t + ϕ o + ϕ e ] + cos [ 2 π ( f D 1 f D 2 ) t + ϕ o ϕ e ] } .
S ( t ) = M cos ( 2 π Δ f D t + Δ ϕ ) .
Δ f D = 2 υ c | ν o ν e | = 2 Δ ν c υ .
υ = c Δ f D 2 Δ ν .
δ υ = ( c Δ f D 2 Δ ν 2 δ Δ ν ) 2 + ( c 2 Δ ν δ Δ f D ) 2 .
δ Δ ν = h l [ ( sin 2 θ n e 2 + cos 2 θ n o 2 ) 1 / 2 n o ] δ ν .
{ ω o τ E = ω o 0 τ E C sin ( ω o τ E arc tan γ ) ω e τ E = ω e 0 τ E C sin ( ω e τ E arc tan γ ) .
δ Δ f D Δ f D Δ n / F s Δ t .

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