Digitally enhanced homodyne interferometry

Andrew J. Sutton; Oliver Gerberding; Gerhard Heinzel; Daniel A. Shaddock

doi:10.1364/OE.20.022195

Digitally enhanced homodyne interferometry

Andrew J. Sutton, Oliver Gerberding, Gerhard Heinzel, and Daniel A. Shaddock

Optics Express
Vol. 20,
Issue 20,
pp. 22195-22207
(2012)
•https://doi.org/10.1364/OE.20.022195

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Andrew J. Sutton, Oliver Gerberding, Gerhard Heinzel, and Daniel A. Shaddock, "Digitally enhanced homodyne interferometry," Opt. Express 20, 22195-22207 (2012)

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Related Topics
Table of Contents Category
- Instrumentation, Measurement, and Metrology
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Instrumentation, measurement, and metrology (120.0120)
- Homodyning (120.2920)
- Interferometry (120.3180)
- Metrology (120.3940)
- Optical instruments (120.4640)
- Phase measurement (120.5050)
- Phase modulation (120.5060)
- Optical sensing and sensors (280.4788)

About this Article
History
- Original Manuscript: July 25, 2012
- Revised Manuscript: August 29, 2012
- Manuscript Accepted: August 30, 2012
- Published: September 13, 2012

Accessible

Open Access

Abstract

We present two variations of a novel interferometry technique capable of simultaneously measuring multiple targets with high sensitivity. The technique performs a homodyne phase measurement by application of a four point phase shifting algorithm, with pseudo-random switching between points to allow multiplexed measurement based upon propagation delay alone. By multiplexing measurements and shifting complexity into signal processing, both variants realise significant complexity reductions over comparable methods. The first variant performs a typical coherent detection with a dedicated reference field and achieves a displacement noise floor 0.8pm/√Hz above 50Hz. The second allows for removal of the dedicated reference, resulting in further simplifications and improved low frequency performance with a 1pm/√Hz noise floor measured down to 20Hz. These results represent the most sensitive measurement performed using this style of interferometry whilst simultaneously reducing the electro-optic footprint.

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References

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Publication

The LIGO Scientific Collaboration, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Rep. Prog. Phys. 72, 076901 (2009).
T. T.-Y. Lam, M. B. Gray, D. A. Shaddock, D. E. McClelland, and J. H. Chow, “Subfrequency noise signal extraction in fiber-optic strain sensors using postprocessing,” Opt. Lett. 37, 2169–2171 (2012).
[Crossref]
B. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. Klipstein, and W. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J Geodesy 10.1007/s00190-012-0566-3 (2012).
[Crossref]
M. Evans, N. Mavalvala, P. Fritschel, R. Bork, B. Bhawal, R. Gustafson, W. Kells, M. Landry, D. Sigg, R. Weiss, S. Whitcomb, and H. Yamamoto, “Lock acquisition of a gravitational-wave interferometer,” Opt. Lett. 27, 598– 600 (2002).
[Crossref]
A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).
N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[Crossref]
G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the Laser Interferometer Space Antenna,” Phys. Rev. Lett. 104, 1–4 (2010).
[Crossref]
S. Anza, M. Armano, E. Balaguer, M. Benedetti, C. Boatella, P. Bosetti, D. Bortoluzzi, N. Brandt, C. Braxmaier, M. Caldwell, L. Carbone, A. Cavalleri, A. Ciccolella, I. Cristofolini, M. Cruise, M. Da Lio, K. Danzmann, D. Desiderio, R. Dolesi, N. Dunbar, W. Fichter, C. Garcia, E. Garcia-Berro, A. F. Garcia Marin, R. Gerndt, A. Gianolio, D. Giardini, R. Gruenagel, A. Hammesfahr, G. Heinzel, J. Hough, D. Hoyl, M. Hueller, O. Jennrich, U. Johann, S. Kemble, C. Killow, D. Kolbe, M. Landgraf, A. Lobo, V. Lorizzo, D. Mance, K. Middleton, F. Nappo, M. Nofrarias, G. Racca, J. Ramos, D. Robertson, M. Sallusti, M. Sandford, J. Sanjuan, P. Sarra, A. Selig, D. Shaul, D. Smart, M. Smit, L. Stagnaro, T. Sumner, C. Tirabassi, S. Tobin, S. Vitale, V. Wand, H. Ward, W. J. Weber, and P. Zweifel, “The LTP experiment on the LISA Pathfinder mission,” Classical Quant. Grav. 22, S125 (2005).
[Crossref]
D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett. 32, 3355–3357 (2007).
[Crossref]
S. Golomb, “On the classification of balanced binary sequences of period 2n-1(Corresp.),” IEEE Trans. Inform. Theory 26, 730–732 (1980).
[Crossref]
S. Raghavan, M. Shane, and R. Yowell, “Spread spectrum codes for GPS L5,” in 2006 Aerosp. Conf. Proc., 1–7 (2006).
[Crossref]
N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation CW lidar,” Appl. Opt. 22, 1382–1386 (1983).
[Crossref]
G. de Vine, D. S. Rabeling, B. J. J. Slagmolen, T. T.-Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. a. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express 17, 828–837 (2009).
[Crossref]
D. M. R. Wuchenich, T. T.-Y. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36, 672–674 (2011).
[Crossref]
O. P. Lay, S. Dubovitsky, D. A. Shaddock, and B. Ware, “Coherent range-gated laser displacement metrology with compact optical head,” Opt. Lett. 32, 2933–2935 (2007).
[Crossref]
K. Dahl, G. Heinzel, B. Willke, K. A. Strain, S. Goler, and K. Danzmann, “Suspension platform interferometer for the AEI 10 m prototype: concept, design and optical layout,” Class. Quantum Grav. 29095024 (2012).
[Crossref]
O. Jennrich, “LISA technology and instrumentation,” Class. Quantum Grav. 26153001 (2009).
[Crossref]
D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]
I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]
Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[Crossref]
Y. Surrel, “Fringe analysis,” Top. Appl. Phys. 77/2000, 55–102 (2000).
[Crossref]
E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref]
G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]
P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.5–7 (1987).
J. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994).
P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006).
[Crossref]
M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol. 27, 901–914 (2009).
[Crossref]
P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981).
[Crossref]
A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[Crossref]
W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]
T. G. McRae, M. T. L. Hsu, C. H. Freund, D. A. Shaddock, J. Herrmann, and M. B. Gray, “Linearization and minimization of cyclic error with heterodyne laser interferometry,” Opt. Lett. 37, 2448–2450 (2012).
[Crossref]

2012 (4)

A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

K. Dahl, G. Heinzel, B. Willke, K. A. Strain, S. Goler, and K. Danzmann, “Suspension platform interferometer for the AEI 10 m prototype: concept, design and optical layout,” Class. Quantum Grav. 29095024 (2012).
[Crossref]

T. T.-Y. Lam, M. B. Gray, D. A. Shaddock, D. E. McClelland, and J. H. Chow, “Subfrequency noise signal extraction in fiber-optic strain sensors using postprocessing,” Opt. Lett. 37, 2169–2171 (2012).
[Crossref]

T. G. McRae, M. T. L. Hsu, C. H. Freund, D. A. Shaddock, J. Herrmann, and M. B. Gray, “Linearization and minimization of cyclic error with heterodyne laser interferometry,” Opt. Lett. 37, 2448–2450 (2012).
[Crossref]

2011 (2)

D. M. R. Wuchenich, T. T.-Y. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36, 672–674 (2011).
[Crossref]

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref]

2010 (2)

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the Laser Interferometer Space Antenna,” Phys. Rev. Lett. 104, 1–4 (2010).
[Crossref]

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

2009 (5)

G. de Vine, D. S. Rabeling, B. J. J. Slagmolen, T. T.-Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. a. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express 17, 828–837 (2009).
[Crossref]

O. Jennrich, “LISA technology and instrumentation,” Class. Quantum Grav. 26153001 (2009).
[Crossref]

The LIGO Scientific Collaboration, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Rep. Prog. Phys. 72, 076901 (2009).

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol. 27, 901–914 (2009).
[Crossref]

2007 (2)

O. P. Lay, S. Dubovitsky, D. A. Shaddock, and B. Ware, “Coherent range-gated laser displacement metrology with compact optical head,” Opt. Lett. 32, 2933–2935 (2007).
[Crossref]

D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett. 32, 3355–3357 (2007).
[Crossref]

2006 (3)

P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006).
[Crossref]

S. Raghavan, M. Shane, and R. Yowell, “Spread spectrum codes for GPS L5,” in 2006 Aerosp. Conf. Proc., 1–7 (2006).
[Crossref]

D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

2005 (1)

S. Anza, M. Armano, E. Balaguer, M. Benedetti, C. Boatella, P. Bosetti, D. Bortoluzzi, N. Brandt, C. Braxmaier, M. Caldwell, L. Carbone, A. Cavalleri, A. Ciccolella, I. Cristofolini, M. Cruise, M. Da Lio, K. Danzmann, D. Desiderio, R. Dolesi, N. Dunbar, W. Fichter, C. Garcia, E. Garcia-Berro, A. F. Garcia Marin, R. Gerndt, A. Gianolio, D. Giardini, R. Gruenagel, A. Hammesfahr, G. Heinzel, J. Hough, D. Hoyl, M. Hueller, O. Jennrich, U. Johann, S. Kemble, C. Killow, D. Kolbe, M. Landgraf, A. Lobo, V. Lorizzo, D. Mance, K. Middleton, F. Nappo, M. Nofrarias, G. Racca, J. Ramos, D. Robertson, M. Sallusti, M. Sandford, J. Sanjuan, P. Sarra, A. Selig, D. Shaul, D. Smart, M. Smit, L. Stagnaro, T. Sumner, C. Tirabassi, S. Tobin, S. Vitale, V. Wand, H. Ward, W. J. Weber, and P. Zweifel, “The LTP experiment on the LISA Pathfinder mission,” Classical Quant. Grav. 22, S125 (2005).
[Crossref]

2002 (1)

M. Evans, N. Mavalvala, P. Fritschel, R. Bork, B. Bhawal, R. Gustafson, W. Kells, M. Landry, D. Sigg, R. Weiss, S. Whitcomb, and H. Yamamoto, “Lock acquisition of a gravitational-wave interferometer,” Opt. Lett. 27, 598– 600 (2002).
[Crossref]

2000 (1)

Y. Surrel, “Fringe analysis,” Top. Appl. Phys. 77/2000, 55–102 (2000).
[Crossref]

1996 (1)

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[Crossref]

1993 (1)

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

1992 (1)

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]

1990 (1)

A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[Crossref]

1987 (1)

P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.5–7 (1987).

1983 (1)

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation CW lidar,” Appl. Opt. 22, 1382–1386 (1983).
[Crossref]

1981 (1)

P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981).
[Crossref]

1980 (1)

S. Golomb, “On the classification of balanced binary sequences of period 2n-1(Corresp.),” IEEE Trans. Inform. Theory 26, 730–732 (1980).
[Crossref]

Acton, D.

A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

Anza, S.

Armano, M.

Baba, H.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation CW lidar,” Appl. Opt. 22, 1382–1386 (1983).
[Crossref]

Balaguer, E.

Benedetti, M.

Bhawal, B.

Boatella, C.

Bobroff, N.

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

A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[Crossref]

Bork, R.

Bortoluzzi, D.

Bosetti, P.

Bouchez, A.

A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

Brandt, N.

Braxmaier, C.

Burke, J.

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref]

Bykov, I.

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]

Caldwell, M.

Carbone, L.

Cavalleri, A.

Cho, P. S.

P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006).
[Crossref]

Chow, J. H.

Chua, S.

Ciccolella, A.

Cristofolini, I.

Cruise, M.

Da Lio, M.

Dahl, K.

Danzmann, K.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]

B. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. Klipstein, and W. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J Geodesy 10.1007/s00190-012-0566-3 (2012).
[Crossref]

de Vine, G.

Delgado, J. J. E.

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]

Desiderio, D.

Dolesi, R.

Dubovitsky, S.

O. P. Lay, S. Dubovitsky, D. A. Shaddock, and B. Ware, “Coherent range-gated laser displacement metrology with compact optical head,” Opt. Lett. 32, 2933–2935 (2007).
[Crossref]

Dunbar, N.

Evans, M.

Feng, W.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

Fichter, W.

Folkner, W.

Freund, C. H.

Fritschel, P.

Garcia, C.

Garcia Marin, A. F.

García Marin, A. F.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

Garcia-Berro, E.

Gerndt, R.

Gianolio, A.

Giardini, D.

Goler, S.

Golomb, S.

S. Golomb, “On the classification of balanced binary sequences of period 2n-1(Corresp.),” IEEE Trans. Inform. Theory 26, 730–732 (1980).
[Crossref]

Gray, M. B.

Gruenagel, R.

Gustafson, R.

Guzmán Cervantes, F.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

Hack, E.

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref]

Halverson, P.

D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Hammesfahr, A.

Hariharan, P.

P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.5–7 (1987).

Heinzel, G.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]

Herrmann, J.

Heydemann, P. L. M.

P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981).
[Crossref]

Hou, W.

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]

Hough, J.

Hoyl, D.

Hsu, M. T. L.

Hueller, M.

Jennrich, O.

O. Jennrich, “LISA technology and instrumentation,” Class. Quantum Grav. 26153001 (2009).
[Crossref]

Johann, U.

Kanneganti, S.

A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

Kells, W.

Kemble, S.

Khurgin, J. B.

P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006).
[Crossref]

Killow, C.

Klipstein, B.

D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Klipstein, W.

Klipstein, W. M.

Kolbe, D.

Kullmann, J.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010).
[Crossref]

Lam, T. T.-Y.

Landgraf, M.

Landry, M.

Lay, O. P.

O. P. Lay, S. Dubovitsky, D. A. Shaddock, and B. Ware, “Coherent range-gated laser displacement metrology with compact optical head,” Opt. Lett. 32, 2933–2935 (2007).
[Crossref]

Lobo, A.

Lorizzo, V.

Mance, D.

Marín, A. F. G.

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009).
[Crossref]

Mavalvala, N.

McClelland, D. E.

McKenzie, K.

McLeod, B.

A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

McRae, T. G.

Middleton, K.

Nappo, F.

Nofrarias, M.

Oreb, B.

P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.5–7 (1987).

Proakis, J.

J. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994).

Rabeling, D. S.

Racca, G.

Raghavan, S.

S. Raghavan, M. Shane, and R. Yowell, “Spread spectrum codes for GPS L5,” in 2006 Aerosp. Conf. Proc., 1–7 (2006).
[Crossref]

Ramos, J.

Robertson, D.

Rosenbluth, A. E.

A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[Crossref]

Sakurai, K.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation CW lidar,” Appl. Opt. 22, 1382–1386 (1983).
[Crossref]

Salehi, M.

J. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994).

Sallusti, M.

Sandford, M.

Sanjuan, J.

Sarra, P.

Selig, A.

Shaddock, D.

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Fig. 1 (a) shows the interference fringe (dotted line) F = cos(φ − φ_S) for a phase offset φ and time-varying phase shift φ_S. A heterodyne offset frequency Δf produces a linear phase shift φ_S(t)= 2π Δft, mapping out the fringe over T = 1/Δf sec. Also shown are the cosine in-phase (I) and sine quadrature (Q) decompositions used for phase estimation. Discrete phase steps of φ_S(t) = {0,π /2,π, 3π /2} rad will result in the phase stepped (solid) response which directly samples the I and Q amplitudes. (b) Depicts the complex phasor “symbol” for each phase step resulting from the modulation sequences c_I[n] and c_Q[n].

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Fig. 2 Showing a proposed fibre fibre ring interferometer setup. The laser is split into LO and metrology branches. The Metrology branch is modulated with I and Q PRN sequences using a Dual Parallel Mach-Zender modulator. A small (∼ 5%) pick-off to detector PD_C is used for modulator control. The system under test is a fibre ring, formed using a 50/50 coupler with a fibre connected EOM for calibration signal injection. The ring output is interfered with the LO at detector PD_S for detection. The ring will create a set of distinct ‘round-trip’ signals S_i, with decaying amplitude due to losses, which will be separable based upon different propagation delay τ_i. This is illustrated by the ring impulse response, which plots signal strength against propagation delay.

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Fig. 3 Showing the measured impulse response of the interferometer by correlating the measured output against delayed versions of the modulation sequence. The fibre ring round trips can be seen beginning at ∼ 41ns with 25.6ns spacing between arrival times.

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Fig. 4 (a) showing the raw time domain displacement measurement of the first 4 arriving signals. A 90 Hz calibration tone with single pass modulation depth of ∼ 240nm_pp was applied. Each round trip accumulates phase proportional to the number of passes through the modulator. (b) shows the Root Power Spectral Density (RPSD) for isolated measurements of the first, second and third round trips of the fibre ring. Each round trip senses a single pass of the calibration signal along with 50Hz mains pick-up. Harmonics of the calibration frequency indicate a cyclic error of ∼ 7nm. The difference between round trips #1 and #2 is shown to demonstrate the limited cancellation resulting from cyclic error.

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Fig. 5 Showing the free running displacement RPSD. A picometer scale white noise floor reached above 50Hz. Low frequencies are dominated by incomplete cancellation of fibre acoustic noise, as demonstrated by the Straight through pass.

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Fig. 6 Showing the fibre ring interferometer setup without dedicated Local Oscillator. The Metrology branch is identically prepared and interrogates the same fibre ring as Fig. 2. The largest signal component (lowest loss) results from the straight through pass of the loop coupler. This component may now be used as a LO for the demodulation of the other loop round trip components.

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Fig. 7 Shows the displacement RPSD of the double demodulation measurements for the first, second and third round trips of the fibre ring. A picometer scale white noise floor is measured for all channels above above 3 Hz while low frequencies are dominated by laser frequency noise as predicted by the Expected Laser trace.

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Tables (1)

Table 1 System parameters for the DEHoI implementations with (Calibration) and without (Free-running) the ring EOM.

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Equations (10)

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P (t, n) = {‖ {\tilde{E}}_{I N} (t) ‖}^{2} + {‖ {\tilde{E}}_{L O} (t) ‖}^{2} + \frac{{\tilde{E}}_{L O} (t) {\tilde{E}}_{I N} (t)}{\sqrt{2}} (c_{I} [n] cos (φ_{d}) + c_{Q} [n] sin (φ_{d}))

\begin{array}{l} \hat{I} & = s [1] + s [4] - s [2] - s [3] \approx & 2 \sqrt{2} {\tilde{E}}_{L o} {\tilde{E}}_{I N} cos (φ_{d}) \\ \hat{Q} & = s [1] + s [2] - s [3] - s [4] \approx & 2 \sqrt{2} {\tilde{E}}_{L o} {\tilde{E}}_{I N} sin (φ_{d}) \end{array}} \Rightarrow {\hat{φ}}_{d} = arctan (\frac{\hat{Q}}{\hat{I}})

{\hat{φ}}_{i} (t) = φ_{L} (t - τ_{i} - τ_{R}) + φ_{i} (t - τ_{R}) + φ_{R} (t) - φ_{L o} (t - τ_{L O})

{\tilde{P}}_{Int} (t) \approx \sum_{i = 0}^{N} \frac{α_{i}}{\sqrt{2}} {\tilde{E}}_{L O} (t) {\tilde{E}}_{I N} (t - τ_{i}) [c_{I} (t - τ_{i}) cos {\tilde{φ}}_{i} (t) + c_{Q} (t - τ_{i}) sin {\tilde{φ}}_{i} (t)]

\begin{array}{l} {\hat{x}}_{j} (t) & = & \frac{{\tilde{P}}_{Int} (t) ★ c_{I} (t - τ_{j})}{T_{c}} \\ \approx & \frac{1}{T_{c}} \int_{t - T_{C}}^{t} c_{I} (t - τ_{j}) \sum_{i = 0}^{N_{S}} β_{i} (t) [c_{I} (t - τ_{i}) cos {\tilde{φ}}_{i} (t) + c_{Q} (t - τ_{i}) sin {\tilde{φ}}_{i} (t)] d t \\ \approx & \frac{1}{T_{c}} \int_{t - T_{c}}^{t} β_{j} (t) cos {\hat{φ}}_{j} (t) + β_{j} (t) c_{I} (t - τ_{j}) c_{Q} (t - τ_{j}) sin {\tilde{φ}}_{j} (t) \\ + c_{I} (t - τ_{j}) \sum_{i \neq j} β_{i} (t) [c_{I} (t - τ_{i}) cos {\tilde{φ}}_{i} (t) + c_{Q} (t - τ_{i}) sin {\tilde{φ}}_{i} (t)] d t \end{array}

\begin{array}{l} {\tilde{x}}_{j} (t) & \approx & β_{j} (t) cos {\tilde{φ}}_{j} (t) + β_{j} (t) R_{j, j} sin {\tilde{φ}}_{j} (t) + \sum_{i \neq j} β_{i} (t) [A_{j, i} cos {\tilde{φ}}_{i} (t) + R_{j, i} sin {\tilde{φ}}_{i} (t)] \\ \approx & β_{j} (t) cos {\tilde{φ}}_{j} (t) \end{array}

\begin{array}{l} {\tilde{y}}_{j} (t) & \approx & β_{j} (t) sin {\tilde{φ}}_{j} (t) + β_{j} (t) R_{j, j} cos {\tilde{φ}}_{j} (t) + \sum_{i \neq j} β_{i} (t) [R_{j, i} cos {\tilde{φ}}_{i} (t) + A_{j, i} sin {\tilde{φ}}_{i} (t)] \\ \approx & β_{j} (t) sin {\tilde{φ}}_{j} (t) \end{array}

\begin{array}{l} {\tilde{P}}_{Int} (t) & \approx & \sum_{i = 0}^{N} \sum_{j \neq i}^{N} \frac{α_{i} α_{j}}{2} {\tilde{E}}_{I N} (t - τ_{i}) {\tilde{E}}_{I N} (t - τ_{j}) [c_{I} (t - τ_{i}) c_{I} (t - τ_{j}) cos ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) \\ + c_{Q} (t - τ_{i}) c_{Q} (t - τ_{j}) cos ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) + c_{I} (t - τ_{i}) c_{Q} (t - τ_{j}) sin ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) \\ - c_{Q} (t - τ_{i}) c_{I} (t - τ_{j}) sin ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t))] \end{array}

\begin{array}{l} {I I}_{{\tilde{x}}_{i, j}} (t) & = \frac{1}{T_{c}} ({\tilde{P}}_{Int} (t) ★ (C_{I} (t - τ_{j}) c_{I} (t - τ_{j}))) \approx & β_{i, j} (t) cos ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) \\ Q Q_{{\tilde{x}}_{i, j}} (t) & = \frac{1}{T_{c}} ({\tilde{P}}_{Int} (t) ★ (C_{Q} (t - τ_{j}) c_{Q} (t - τ_{j}))) \approx & β_{i, j} (t) cos ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) \\ I Q_{{\tilde{y}}_{i, j}} (t) & = \frac{1}{T_{c}} ({\tilde{P}}_{Int} (t) ★ (c_{I} (t - τ_{j}) c_{Q} (t - τ_{j}))) \approx & β_{i, j} (t) sin ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) \\ Q I_{{\tilde{y}}_{i, j}} (t) & = \frac{1}{T_{c}} ({\tilde{P}}_{Int} (t) ★ (c_{Q} (t - τ_{j}) c_{I} (t - τ_{j}))) \approx & - β_{i, j} (t) sin ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) \end{array}

\begin{array}{l} {\tilde{p}}_{i, j} (t) & = & ({}^{II}{\tilde{x}}_{i, j} (t) + {}^{QQ}{\tilde{x}}_{i, j} (t)) + i ({}^{IQ}{\tilde{y}}_{i, j} (t) - {}^{QI}{\tilde{y}}_{i, j} (t)) \\ = & 2 β_{i, j} (t) (cos ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t)) + i sin ({\tilde{φ}}_{i} (t) - {\tilde{φ}}_{j} (t))) \end{array}

Parameter	Value

PRN Frequency	312.5 MHz
PRN Code Length	2¹⁵ − 1 points
PRN Code Repetition Rate	9.5 kHz
ADC Frequency	1.25 GSps
Fibre Ring (Calibration)	5 m
Fibre Ring (Free-running)	2 m
Phase Estimate Update Rate	∼ 10kHz