Abstract

An absolute gravimeter is based on interferometric displacement measurement during repeated free falls of a target mirror in a high vacuum. A suitable homodyne quadrature laser interferometer is developed and reported. Standard deviation of the mean absolute gravity value in a typical two-day measurement session is less than 1×108m/s2. Additionally, digital demodulation of the interference signal using a 90° phase-shift filter based on the Hilbert transform is presented. A combination of optical and digital quadrature phase shifts is shown to be helpful to improve accuracy in homodyne interferometers with the accelerated target mirror.

© 2014 Optical Society of America

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  7. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44, 3492–3498 (2005).
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  8. P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
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  9. T. Požar and J. Možina, “Enhanced ellipse fitting in a two-detector homodyne quadrature laser interferometer,” Meas. Sci. Technol. 22, 085301 (2011).
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  30. A. Araya, H. Sakai, Y. Tamura, T. Tsubokawa, and S. Svitlov, “Development of a compact absolute gravimeter with a built-in accelerometer and a silent drop mechanism,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.
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    [CrossRef]
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    [CrossRef]
  39. Q. Sun, W. Wabinski, and T. Bruns, “Investigation of primary vibration calibration at high frequencies using the homodyne quadrature sine-approximation method: problems and solutions,” Meas. Sci. Technol. 17, 2197–2205 (2006).
    [CrossRef]

2013 (1)

2012 (3)

S. Svitlov, “Frequency domain analysis of absolute gravimeters,” Metrologia 49, 706–726 (2012).
[CrossRef]

S. Rerucha, Z. Buchta, M. Sarbort, J. Lazar, and O. Cip, “Detection of interference phase by digital computation of quadrature signals in homodyne laser interferometry,” Sensors 12, 14095–14112 (2012).
[CrossRef]

S. Svitlov, Ch. Rothleitner, and L. J. Wang, “Accuracy assessment of the two-sample zero-crossing detection in a sinusoidal signal,” Metrologia 49, 413–424 (2012).
[CrossRef]

2011 (2)

T. Požar and J. Možina, “Enhanced ellipse fitting in a two-detector homodyne quadrature laser interferometer,” Meas. Sci. Technol. 22, 085301 (2011).
[CrossRef]

J. Lee, H. Yoon, and T. H. Yoon, “High-resolution parallel multipass laser interferometer with an interference fringe spacing of 15  nm,” Opt. Commun. 284, 1118–1122 (2011).

2010 (1)

S. Svitlov, P. Masłyk, Ch. Rothleitner, H. Hu, and L. J. Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia 47, 677–689 (2010).
[CrossRef]

2009 (2)

2008 (2)

M. Pisani, “Multiple reflection Michelson interferometer with picometer resolution,” Opt. Express 16, 21558–21563 (2008).
[CrossRef]

M. Harker, P. O’Leary, and P. Zsombor-Murray, “Direct type-specific conic fitting and eigenvalue bias correction,” Image Vis. Comput. 26, 372–381 (2008).
[CrossRef]

2007 (2)

S. Pullteap, H. C. Seat, and T. Bosch, “Modified fringe-counting technique applied to a dual-cavity fiber Fabry–Pérot vibrometer,” Opt. Eng. 46, 115603 (2007).
[CrossRef]

T.-J. Ahn and D. Y. Kim, “Analysis of nonlinear frequency sweep in high-speed tunable laser sources using a self-homodyne measurement and Hilbert transformation,” Appl. Opt. 46, 2394–2400 (2007).
[CrossRef]

2006 (2)

T. M. Niebauer, A. Schiel, and D. van Westrum, “Complex heterodyne for undersampled chirped sinusoidal signals,” Appl. Opt. 45, 8322–8330 (2006).
[CrossRef]

Q. Sun, W. Wabinski, and T. Bruns, “Investigation of primary vibration calibration at high frequencies using the homodyne quadrature sine-approximation method: problems and solutions,” Meas. Sci. Technol. 17, 2197–2205 (2006).
[CrossRef]

2005 (2)

T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44, 3492–3498 (2005).
[CrossRef]

G. D’Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, “A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry,” Metrologia 42, 233–238 (2005).
[CrossRef]

2004 (2)

2003 (1)

G. Durando, G. Mana, and F. Mazzoleni, “Accuracy assessment of data analysis in absolute gravimetry,” IEEE Trans. Instrum. Meas. 52, 500–503 (2003).
[CrossRef]

2001 (1)

T. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[CrossRef]

2000 (1)

J. Lawall and E. Kessler, “Michelson interferometry with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[CrossRef]

1999 (1)

T. Tsubokawa and S. Svitlov, “New method of digital fringe signal processing in an absolute gravimeter,” IEEE Trans. Instrum. Meas. 48, 488–491 (1999).
[CrossRef]

1996 (2)

H. Hanada, T. Tsubokawa, and S. Tsuruta, “Possible large systematic error source in absolute gravimetry,” Metrologia 33, 155–160 (1996).
[CrossRef]

C. Wu, C. Su, and G. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

1995 (2)

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995).
[CrossRef]

P. R. Parker, M. A. Zumberge, and R. L. Parker, “A new method for fringe-signal processing in absolute gravity meters,” Manuscripta Geodaetica 20, 173–181 (1995).

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

1990 (1)

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).

1984 (1)

T. Tsubokawa, “A fringe signal processing method for an absolute gravimeter,” Metrologia 20, 107–113 (1984).
[CrossRef]

1982 (1)

M. A. Zumberge, R. L. Rinker, and J. E. Faller, “A portable apparatus for absolute measurements of the Earth’s gravity,” Metrologia 18, 145–152 (1982).
[CrossRef]

1981 (1)

1979 (1)

E. H. Ewitt and R. H. Ewitt, “The Gibbs-Wilbraham phenomenon: an episode in Fourier analysis,” Hist. Exact Sci. 21, 129–160 (1979).

Ahn, T.-J.

Araya, A.

S. Svitlov, A. Araya, and T. Tsubokawa, “Digital fringe signal processing methods in absolute gravimetry,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

A. Araya, H. Sakai, Y. Tamura, T. Tsubokawa, and S. Svitlov, “Development of a compact absolute gravimeter with a built-in accelerometer and a silent drop mechanism,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

Barbato, G.

G. D’Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, “A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry,” Metrologia 42, 233–238 (2005).
[CrossRef]

Berger, J.

Birch, K. P.

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Bosch, T.

S. Pullteap, H. C. Seat, and T. Bosch, “Modified fringe-counting technique applied to a dual-cavity fiber Fabry–Pérot vibrometer,” Opt. Eng. 46, 115603 (2007).
[CrossRef]

Bruns, T.

Q. Sun, W. Wabinski, and T. Bruns, “Investigation of primary vibration calibration at high frequencies using the homodyne quadrature sine-approximation method: problems and solutions,” Meas. Sci. Technol. 17, 2197–2205 (2006).
[CrossRef]

Buchta, Z.

S. Rerucha, Z. Buchta, M. Sarbort, J. Lazar, and O. Cip, “Detection of interference phase by digital computation of quadrature signals in homodyne laser interferometry,” Sensors 12, 14095–14112 (2012).
[CrossRef]

Cip, O.

S. Rerucha, Z. Buchta, M. Sarbort, J. Lazar, and O. Cip, “Detection of interference phase by digital computation of quadrature signals in homodyne laser interferometry,” Sensors 12, 14095–14112 (2012).
[CrossRef]

D’Agostino, G.

G. D’Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, “A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry,” Metrologia 42, 233–238 (2005).
[CrossRef]

Desogus, S.

G. D’Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, “A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry,” Metrologia 42, 233–238 (2005).
[CrossRef]

Durando, G.

G. Durando, G. Mana, and F. Mazzoleni, “Accuracy assessment of data analysis in absolute gravimetry,” IEEE Trans. Instrum. Meas. 52, 500–503 (2003).
[CrossRef]

Dzieciuch, M. A.

Eom, T.

T. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[CrossRef]

Ewitt, E. H.

E. H. Ewitt and R. H. Ewitt, “The Gibbs-Wilbraham phenomenon: an episode in Fourier analysis,” Hist. Exact Sci. 21, 129–160 (1979).

Ewitt, R. H.

E. H. Ewitt and R. H. Ewitt, “The Gibbs-Wilbraham phenomenon: an episode in Fourier analysis,” Hist. Exact Sci. 21, 129–160 (1979).

Faller, J. E.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995).
[CrossRef]

M. A. Zumberge, R. L. Rinker, and J. E. Faller, “A portable apparatus for absolute measurements of the Earth’s gravity,” Metrologia 18, 145–152 (1982).
[CrossRef]

Germak, A.

G. D’Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, “A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry,” Metrologia 42, 233–238 (2005).
[CrossRef]

Gonda, S.

Gregorcic, P.

Hanada, H.

H. Hanada, T. Tsubokawa, and S. Tsuruta, “Possible large systematic error source in absolute gravimetry,” Metrologia 33, 155–160 (1996).
[CrossRef]

Harker, M.

M. Harker, P. O’Leary, and P. Zsombor-Murray, “Direct type-specific conic fitting and eigenvalue bias correction,” Image Vis. Comput. 26, 372–381 (2008).
[CrossRef]

Heydemann, P. L. M.

Hilt, R.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995).
[CrossRef]

Hu, H.

S. Svitlov, P. Masłyk, Ch. Rothleitner, H. Hu, and L. J. Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia 47, 677–689 (2010).
[CrossRef]

H. Hu, X. Qiu, J. Wang, A. Ju, and Y. Zhang, “Subdivision and direction recognition of λ/16 of orthogonal fringes for nanometric measurement,” Appl. Opt. 48, 6479–6484 (2009).
[CrossRef]

Huang, Q.

Jeong, K.

T. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[CrossRef]

Ju, A.

Keem, T.

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometry with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[CrossRef]

Kim, D. Y.

Kim, J. Y.

T. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[CrossRef]

Klopping, F.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995).
[CrossRef]

Kurosawa, T.

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometry with 10  pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[CrossRef]

Lazar, J.

S. Rerucha, Z. Buchta, M. Sarbort, J. Lazar, and O. Cip, “Detection of interference phase by digital computation of quadrature signals in homodyne laser interferometry,” Sensors 12, 14095–14112 (2012).
[CrossRef]

Lee, J.

J. Lee, H. Yoon, and T. H. Yoon, “High-resolution parallel multipass laser interferometer with an interference fringe spacing of 15  nm,” Opt. Commun. 284, 1118–1122 (2011).

Mana, G.

G. Durando, G. Mana, and F. Mazzoleni, “Accuracy assessment of data analysis in absolute gravimetry,” IEEE Trans. Instrum. Meas. 52, 500–503 (2003).
[CrossRef]

Maslyk, P.

S. Svitlov, P. Masłyk, Ch. Rothleitner, H. Hu, and L. J. Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia 47, 677–689 (2010).
[CrossRef]

Mazzoleni, F.

G. Durando, G. Mana, and F. Mazzoleni, “Accuracy assessment of data analysis in absolute gravimetry,” IEEE Trans. Instrum. Meas. 52, 500–503 (2003).
[CrossRef]

Misumi, I.

Možina, J.

T. Požar and J. Možina, “Enhanced ellipse fitting in a two-detector homodyne quadrature laser interferometer,” Meas. Sci. Technol. 22, 085301 (2011).
[CrossRef]

P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
[CrossRef]

Niebauer, T. M.

O’Leary, P.

M. Harker, P. O’Leary, and P. Zsombor-Murray, “Direct type-specific conic fitting and eigenvalue bias correction,” Image Vis. Comput. 26, 372–381 (2008).
[CrossRef]

Origlia, C.

G. D’Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, “A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry,” Metrologia 42, 233–238 (2005).
[CrossRef]

Parker, P. R.

P. R. Parker, M. A. Zumberge, and R. L. Parker, “A new method for fringe-signal processing in absolute gravity meters,” Manuscripta Geodaetica 20, 173–181 (1995).

Parker, R. L.

M. A. Zumberge, J. Berger, M. A. Dzieciuch, and R. L. Parker, “Resolving quadrature fringes in real time,” Appl. Opt. 43, 771–775 (2004).
[CrossRef]

P. R. Parker, M. A. Zumberge, and R. L. Parker, “A new method for fringe-signal processing in absolute gravity meters,” Manuscripta Geodaetica 20, 173–181 (1995).

Peng, G.

C. Wu, C. Su, and G. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Pisani, M.

Požar, T.

T. Požar and J. Možina, “Enhanced ellipse fitting in a two-detector homodyne quadrature laser interferometer,” Meas. Sci. Technol. 22, 085301 (2011).
[CrossRef]

P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
[CrossRef]

Pullteap, S.

S. Pullteap, H. C. Seat, and T. Bosch, “Modified fringe-counting technique applied to a dual-cavity fiber Fabry–Pérot vibrometer,” Opt. Eng. 46, 115603 (2007).
[CrossRef]

Qiu, X.

Rerucha, S.

S. Rerucha, Z. Buchta, M. Sarbort, J. Lazar, and O. Cip, “Detection of interference phase by digital computation of quadrature signals in homodyne laser interferometry,” Sensors 12, 14095–14112 (2012).
[CrossRef]

Rinker, R. L.

M. A. Zumberge, R. L. Rinker, and J. E. Faller, “A portable apparatus for absolute measurements of the Earth’s gravity,” Metrologia 18, 145–152 (1982).
[CrossRef]

Rothleitner, Ch.

S. Svitlov, Ch. Rothleitner, and L. J. Wang, “Accuracy assessment of the two-sample zero-crossing detection in a sinusoidal signal,” Metrologia 49, 413–424 (2012).
[CrossRef]

S. Svitlov, P. Masłyk, Ch. Rothleitner, H. Hu, and L. J. Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia 47, 677–689 (2010).
[CrossRef]

Sakai, H.

A. Araya, H. Sakai, Y. Tamura, T. Tsubokawa, and S. Svitlov, “Development of a compact absolute gravimeter with a built-in accelerometer and a silent drop mechanism,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

Sarbort, M.

S. Rerucha, Z. Buchta, M. Sarbort, J. Lazar, and O. Cip, “Detection of interference phase by digital computation of quadrature signals in homodyne laser interferometry,” Sensors 12, 14095–14112 (2012).
[CrossRef]

Sasagawa, G. S.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995).
[CrossRef]

Schiel, A.

Seat, H. C.

S. Pullteap, H. C. Seat, and T. Bosch, “Modified fringe-counting technique applied to a dual-cavity fiber Fabry–Pérot vibrometer,” Opt. Eng. 46, 115603 (2007).
[CrossRef]

Siebert, W. M.

W. M. Siebert, Circuits, Signals and Systems (MIT, 1986).

Su, C.

C. Wu, C. Su, and G. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Sun, Q.

Q. Sun, W. Wabinski, and T. Bruns, “Investigation of primary vibration calibration at high frequencies using the homodyne quadrature sine-approximation method: problems and solutions,” Meas. Sci. Technol. 17, 2197–2205 (2006).
[CrossRef]

Svitlov, S.

S. Svitlov, Ch. Rothleitner, and L. J. Wang, “Accuracy assessment of the two-sample zero-crossing detection in a sinusoidal signal,” Metrologia 49, 413–424 (2012).
[CrossRef]

S. Svitlov, “Frequency domain analysis of absolute gravimeters,” Metrologia 49, 706–726 (2012).
[CrossRef]

S. Svitlov, P. Masłyk, Ch. Rothleitner, H. Hu, and L. J. Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia 47, 677–689 (2010).
[CrossRef]

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[CrossRef]

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S. Svitlov, A. Araya, and T. Tsubokawa, “Digital fringe signal processing methods in absolute gravimetry,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

Tamura, Y.

A. Araya, H. Sakai, Y. Tamura, T. Tsubokawa, and S. Svitlov, “Development of a compact absolute gravimeter with a built-in accelerometer and a silent drop mechanism,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

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J. T. Tristan and R. C. Watson, Digital Demodulation of Interferometric Signals, Modern Metrology Concerns, Dr. L. Cocco, ed. (2012), pp. 323–329, http://www.intechopen.com/books/modern-metrology-concerns/digital-demodulation-of-interferometric-signals .

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A. Araya, H. Sakai, Y. Tamura, T. Tsubokawa, and S. Svitlov, “Development of a compact absolute gravimeter with a built-in accelerometer and a silent drop mechanism,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

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[CrossRef]

S. Svitlov, P. Masłyk, Ch. Rothleitner, H. Hu, and L. J. Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia 47, 677–689 (2010).
[CrossRef]

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A. Araya, H. Sakai, Y. Tamura, T. Tsubokawa, and S. Svitlov, “Development of a compact absolute gravimeter with a built-in accelerometer and a silent drop mechanism,” in International Association of Geodesy (IAG) Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM-2013), Saint Petersburg, Russia, 17–20 September 2013.

J. T. Tristan and R. C. Watson, Digital Demodulation of Interferometric Signals, Modern Metrology Concerns, Dr. L. Cocco, ed. (2012), pp. 323–329, http://www.intechopen.com/books/modern-metrology-concerns/digital-demodulation-of-interferometric-signals .

W. Torge, Gravimetry (Walter de Grueter, 1989), pp. 133–136.

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Figures (8)

Fig. 1.
Fig. 1.

Free-fall absolute gravimeter with the homodyne quadrature interferometer and an algorithm of fringe signals processing on the MATLAB platform: BS1, BS2, beam splitters; PBS, polarizing beam splitter; M, mirror; QWP, quarter-wave plate; HWP, half-wave plate; D0, D1, D2, photodetectors; ADC, analog-to-digital converter; HT, Hilbert transform; LSF, least-squares fitting.

Fig. 2.
Fig. 2.

Quadrature fringe signals U1 and U2 at the start of a free-fall (left) produce a distorted ellipse (right) on the Lissajous pattern (signals are preliminary centered, dots). Corrected signals form a circle (open dots) with a fitted radius of R=0.97V (solid line).

Fig. 3.
Fig. 3.

Amplitude (top) and phase (bottom) frequency responses of the ideal quadrature filter realized using the Hilbert transform.

Fig. 4.
Fig. 4.

Real quadrature filter built using the MATLAB function hilbert.m. (a) Sinusoidal signal (solid line) and its Hilbert transform (open dots), signal frequency is 1/25 of the sampling frequency; total interval T contains 7.06 signal periods. The quadrature errors (b) is for the case (a), while (c) is for the frequency, which sweeps from 0.6 to 4.8 MHz. A truncated time interval as used in the TAG-1 is also shown.

Fig. 5.
Fig. 5.

Gravity measurements with the default processing method. (a) Quadrature error correction computed by Eq. (5). (b) De-tided row gravity data (solid dots) and outliers (crosses), a constant value of 980121000.0 μGal is subtracted. (c) Experimental histogram follows the theoretical Gaussian curve (solid line).

Fig. 6.
Fig. 6.

Single-drop least-squares residuals in the second-difference method, first channel (solid line); digital quadrature, first channel (open dots); optical quadrature, both channels (solid dots).

Fig. 7.
Fig. 7.

Averaged least squares residuals in the second-difference method, first channel (a); digital quadrature method, first channel (b); optical quadrature method, both channels (c). Relevant amplitude spectrum (right) is computed using the MATLAB function fft.m.

Fig. 8.
Fig. 8.

Gravity measurements as processed with four different methods: SDif, second difference method; DQuad, digital quadrature based on the Hilbert transform; SFit, nonlinear least-squares fit of the frequency-swept model of the fringe signal; OQuad, optical quadrature. Dashed lines connect results for the first and second channels, respectively. Error bars represent standard deviation of the mean g.

Tables (1)

Tables Icon

Table 1. Displacement Resolution and Measurement Uncertainty from the Least-Squares Residuals Analysis

Equations (19)

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U1*=RcosΦ;U2*=RsinΦ,
U1=U1*+p,U2=1r(U2*cosαU1*sinα)+q,
(U1p)2+[(U2q)r+(U1p)sinαcosα]2=R2,
AX1+BX2+CX3+DX4+EX5=1,
α=sin1[C(4AB)1/2],
r=(B/A)1/2,
p=2BDECC24AB,
q=2AEDCC24AB,
R=2r(A+Ap2+Bq2+Cpq4ABC2)1/2.
U1*=U1p,
U2*=(U2q)r+(U1p)sinαcosα.
S(t)=λ4πΦ(t)=λ4πtan1U2*(t)U1*(t).
z(ti)=z0+v0ti+12g0ti2+εi,i=1,2,,N,
U˜(t)={U(t)}=1πU(τ)tτdt.
H(jω)={j,ω<00,ω=0j,ω>0.
S(t)=λ4πtan1{U(t)}U(t).
σ=125T2σεN,
σk=Zk2ωk2|A(ωk)|,
A(ω)=120ω5T5[(12ω2T2)sinωT26ωTcosωT2].

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