Abstract

A discrete fringe phase unwrapping algorithm (DFPUA) based on Kalman motion estimation is proposed to accurately demodulate the phases of I/Q-interferometers with deeply under-sampled quadrature signals, thus to break through the limitations of the Nyquist frequency for high-speed measurement. The basic concept of DFPUA is to estimate the current displacement according to the former motion state, then confirm the actual phase integer number by comparing the estimated phase decimal with the actual phase decimal; in this way, peak acceleration/jerk instead of peak velocity becomes the factor that determines the sampling rate. Two types of DFPUA including velocity estimation and velocity-acceleration estimation are illustrated in detail. Simulation experiment results indicate that the DFPUA realizes a significant reduction in the sampling rate and the amount of data for low frequency vibration measurement, proposing a practical approach for high-speed and long-time measurement such as ultra-low frequency vibration calibration.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Effects of random vibration in high-speed phase-shifting speckle pattern interferometry

Pablo D. Ruiz, Jonathan M. Huntley, Yuji Shen, C. Russell Coggrave, and Guillermo H. Kaufmann
Appl. Opt. 41(19) 3941-3949 (2002)

High-speed, sub-Nyquist interferometry

Tao Wu, Jesus D. Valera, and Andrew J. Moore
Opt. Express 19(11) 10111-10123 (2011)

Bandpass-sampling-based heterodyne interferometer signal acquisition for vibration measurements in primary vibration calibration

Ming Yang, Haijiang Zhu, Chenguang Cai, Ying Wang, and Zhihua Liu
Appl. Opt. 57(29) 8586-8592 (2018)

References

  • View by:
  • |
  • |
  • |

  1. P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17(18), 16322–16331 (2009).
    [Crossref] [PubMed]
  2. S. H. Eang and K. Cho, “Balanced-path homodyne I/Q-interferometer scheme with very simple optical arrangement using a polarizing beam displacer,” Opt. Express 25(7), 8237–8244 (2017).
    [Crossref] [PubMed]
  3. S. H. Eang, S. Yoon, J. G. Park, and K. Cho, “Scanning balanced-path homodyne I/Q-interferometer scheme and its applications,” Opt. Lett. 40(11), 2457–2460 (2015).
    [Crossref] [PubMed]
  4. J. G. Park and K. Cho, “High-precision tilt sensor using a folded Mach-Zehnder geometry in-phase and quadrature interferometer,” Appl. Opt. 55(9), 2155–2159 (2016).
    [Crossref] [PubMed]
  5. D. Musinski, “Displacement-measuring interferometers provide precise metrology,” Laser Focus World 39(12), 80 (2003).
  6. T. Bruns and S. Gazioch, “Correction of shaker flatness deviations in very low frequency primary accelerometer calibration,” Metrologia 53(3), 986–990 (2016).
    [Crossref]
  7. W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
    [Crossref]
  8. L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014).
    [Crossref] [PubMed]
  9. X. M. Xie and Q. N. Zeng, “Efficient and robust phase unwrapping algorithm based on unscented Kalman filter, the strategy of quantizing paths-guided map, and pixel classification strategy,” Appl. Opt. 54(31), 9294–9307 (2015).
    [Crossref] [PubMed]
  10. R. Kulkarni and P. Rastogi, “Phase derivative estimation from a single interferogram using a Kalman smoothing algorithm,” Opt. Lett. 40(16), 3794–3797 (2015).
    [Crossref] [PubMed]

2017 (1)

2016 (2)

J. G. Park and K. Cho, “High-precision tilt sensor using a folded Mach-Zehnder geometry in-phase and quadrature interferometer,” Appl. Opt. 55(9), 2155–2159 (2016).
[Crossref] [PubMed]

T. Bruns and S. Gazioch, “Correction of shaker flatness deviations in very low frequency primary accelerometer calibration,” Metrologia 53(3), 986–990 (2016).
[Crossref]

2015 (3)

2014 (2)

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014).
[Crossref] [PubMed]

2009 (1)

2003 (1)

D. Musinski, “Displacement-measuring interferometers provide precise metrology,” Laser Focus World 39(12), 80 (2003).

Bruns, T.

T. Bruns and S. Gazioch, “Correction of shaker flatness deviations in very low frequency primary accelerometer calibration,” Metrologia 53(3), 986–990 (2016).
[Crossref]

Cho, K.

Eang, S. H.

Gazioch, S.

T. Bruns and S. Gazioch, “Correction of shaker flatness deviations in very low frequency primary accelerometer calibration,” Metrologia 53(3), 986–990 (2016).
[Crossref]

Gregorcic, P.

He, W.

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

Kulkarni, R.

Liu, Z.

Možina, J.

Musinski, D.

D. Musinski, “Displacement-measuring interferometers provide precise metrology,” Laser Focus World 39(12), 80 (2003).

Park, J. G.

Požar, T.

Rastogi, P.

Shen, R.

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

Tao, L.

Wang, C.

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

Xie, X. M.

Yoon, S.

Yu, M.

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

Zeng, Q. N.

Zhang, W.

Zhang, X.

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

Zhou, Y.

Appl. Opt. (2)

Laser Focus World (1)

D. Musinski, “Displacement-measuring interferometers provide precise metrology,” Laser Focus World 39(12), 80 (2003).

Meas. Sci. Technol. (1)

W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014).
[Crossref]

Metrologia (1)

T. Bruns and S. Gazioch, “Correction of shaker flatness deviations in very low frequency primary accelerometer calibration,” Metrologia 53(3), 986–990 (2016).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Principle of DFPUA based on velocity estimation (sampling interval: Δ T ).
Fig. 2
Fig. 2 Principle of DFPUA based on velocity-acceleration estimation (sampling interval: Δ T ).
Fig. 3
Fig. 3 Data processing of quadrature signals in the measurement of vibration (fv = 500 Hz, apeak = 10 m/s2) using DFPUA based on velocity estimation (fs = 7955 Hz). (a) Quadrature signals. (b) Phases. (c) Phase integer numbers unwrapped.
Fig. 4
Fig. 4 Unwrapped displacement signals (apeak = 10 m/s2) using DFPUA based on velocity estimation and DFPUA velocity-acceleration estimation. (a) fv = 100 Hz. (b) fv = 1 kHz.
Fig. 5
Fig. 5 The minimum requirement for SPUA and two proposed DFPUAs vs. vibration frequency. (a) Sampling rate requirement. (b) Number of samples per channel.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

d k = λ 4 π ( φ k + m k 2 π ) , m k = 0 , ± 1 , ± 2... φ k = arctan I x k I y k
d ( t + Δ T ) = d ( t ) + v ( t ) Δ T + 1 2 a ( t ) Δ T 2 + 1 3 j ( t ) Δ T 3 + ...
x k + 1 = A x k
A = [ 1 Δ T Δ T 2 / 2 ... 0 1 Δ T ... 0 0 ... ... 0 ... 0 1 ] .
v ˜ k = d k d k 1 Δ T , a ˜ k = v ˜ k v ˜ k 1 Δ T , j ˜ k = a ˜ k a ˜ k 1 Δ T ,
x k + 1 = A ( x ˜ k + ω k )
ω k = [ 0 v k v ˜ k a k a ˜ k j k j ˜ k ... ] T
m k + 1 estimated = r o u n d ( d k + 1 estimated λ / 2 ) , φ k + 1 estimated = r e m ( 4 π d k + 1 estimated λ , 2 π )
m k + 1 = { m k + 1 estimated , when | φ k + 1 φ k + 1 estimated | π m k + 1 estimated 1 , when ( φ k + 1 φ k + 1 estimated ) > π m k + 1 estimated + 1 , when ( φ k + 1 φ k + 1 estimated ) < π
φ = { arctan ( I x / I y ) if I x and I y 0 arctan ( I x / I y ) + π if I x and I y < 0 arctan ( I x / I y ) + 2 π if I x * I y < 0
d 0 = λ 4 π φ 0 d 1 = { λ 4 π φ 1 , if | φ 0 φ 1 | π λ 4 π ( φ 1 + 2 π ) , if ( φ 0 φ 1 ) > π λ 4 π ( φ 1 2 π ) , if ( φ 0 φ 1 ) < π
v ˜ k 1 = d k d k 1 Δ T , d k + 1 estimated = d k + v ˜ k 1 * Δ T m k + 1 estimated = r o u n d ( d k + 1 estimated λ / 2 ) , φ k + 1 estimated = r e m ( 4 π d k + 1 estimated λ , 2 π )
e estimated a p e a k Δ T 2
f s 2 a p e a k λ
f s a p e a k λ / 4 2 e r r o r PV
d 0 = λ 4 π φ 0 d 1 = { λ 4 π φ 1 , if | φ 0 φ 1 | π λ 4 π ( φ 1 + 2 π ) , if ( φ 0 φ 1 ) > π λ 4 π ( φ 1 2 π ) , if ( φ 0 φ 1 ) < π , d 2 = { λ 4 π φ 2 , if | φ 1 φ 2 | π λ 4 π ( φ 2 + 2 π ) , if ( φ 1 φ 2 ) > π λ 4 π ( φ 2 2 π ) , if ( φ 1 φ 2 ) < π
v ˜ k 2 = d k 1 d k 2 Δ T , v ˜ k 1 = d k d k 1 Δ T , a ˜ k 2 = v ˜ k 1 v ˜ k 2 Δ T v ˜ k estimated = v ˜ k 1 + a ˜ k 2 * Δ T , d k + 1 estimated = d k + v ˜ k estimated * Δ T m k + 1 estimated = r o u n d ( d k + 1 estimated λ / 2 ) , φ k + 1 estimated = r e m ( 4 π d k + 1 estimated λ , 2 π )
e estimated j p e a k Δ T 3
f s 4 j p e a k λ 3
f s j p e a k λ / 4 2 e r r o r PV 3
a ( t ) = { 0.25 a p e a k sin ( 2 π f v t ) , 0 t T v 0.5 a p e a k sin ( 2 π f v ( t T v ) ) , T v < t 1.5 T v a p e a k sin ( 2 π f v ( t 1.5 T v ) ) , t > 1.5 T v
v p e a k = a p e a k 2 π f v
I x = cos φ M o d , I y = sin ( φ M o d + α )

Metrics