Abstract

Experiments for space and ground-based gravitational wave detectors often require a large dynamic range interferometric position readout of test masses with 1pm/Hz precision over long time scales. Heterodyne interferometer schemes that achieve such precisions are available, but they require complex optical set-ups, limiting their scalability for multiple channels. This article presents the first experimental results on deep frequency modulation interferometry, a new technique that combines sinusoidal laser frequency modulation in unequal arm length interferometers with a non-linear fit algorithm. We have tested the technique in a Michelson and a Mach-Zehnder Interferometer topology, respectively, demonstrated continuous phase tracking of a moving mirror and achieved a performance equivalent to a displacement sensitivity of 250pm/Hz at 1 mHz between the phase measurements of two photodetectors monitoring the same optical signal. By performing time series fitting of the extracted interference signals, we measured that the linearity of the laser frequency modulation is on the order of 2% for the laser source used.

© 2016 Optical Society of America

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References

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  1. K.-S. Isleif, O. Gerberding, S. Köhlenbeck, A. Sutton, B. Sheard, S. Goßler, D. Shaddock, G. Heinzel, and K. Danzmann, “Highspeed multiplexed heterodyne interferometry,” Opt. Express 22, 24689–24696 (2014).
    [Crossref] [PubMed]
  2. A. J. Sutton, O. Gerberding, G. Heinzel, and D. A. Shaddock, “Digitally enhanced homodyne interferometry,” Opt. Express 20, 22195–22207 (2012).
    [Crossref] [PubMed]
  3. D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett. 32, 3355–3357 (2007).
    [Crossref] [PubMed]
  4. 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] [PubMed]
  5. 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] [PubMed]
  6. T. S. Schwarze, O. Gerberding, F. G. Cervantes, G. Heinzel, and K. Danzmann, “Advanced phasemeter for deep phase modulation interferometry,” Opt. Express 22, 18214–18223 (2014).
    [Crossref] [PubMed]
  7. K. Danzmann and et al., “The Gravitational Universe: Whitepaper for the ESA L2/L3 selection,” (2013).
  8. G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
    [Crossref]
  9. M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.
  10. O. Gerberding, “Deep frequency modulation interferometry,” Opt. Express 23, 14753–14762 (2015).
    [Crossref] [PubMed]
  11. J. Zheng, Optical frequency-modulated continuous-wave (FMCW) interferometry (Springer Science & Business Media, 2005), Vol. 107.
  12. I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987).
    [Crossref]
  13. T. Kissinger, T. O. Charrett, and R. P. Tatam, “Range-resolved interferometric signal processing using sinusoidal optical frequency modulation,” Opt. Express 23, 9415–9431 (2015).
    [Crossref] [PubMed]
  14. R. Fleddermann, “Interferometry for a space-based gravitational wave observatory reciprocity of an optical fiber,” Ph.D. thesis, Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz UniversitätHannover (2012).
  15. M. Dehne, M. Tröbs, G. Heinzel, and K. Danzmann, “Verification of polarising optics for the LISA optical bench,” Opt. Express 20, 27273–27287 (2012).
    [Crossref] [PubMed]
  16. O. Gerberding and et al., “Readout for intersatellite laser interferometry: Measuring low frequency phase fluctuations of high-frequency signals with microradian precision,” Rev. Sci. Instrum. 86, 074501 (2015).
    [Crossref] [PubMed]

2015 (3)

2014 (2)

2012 (2)

2010 (1)

2009 (1)

2007 (1)

2004 (1)

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

1987 (1)

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987).
[Crossref]

Braxmaier, C.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Cervantes, F. G.

Charrett, T. O.

Chua, S.

Danzmann, K.

de Vine, G.

Dehne, M.

Drinkwater, M. R.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

Fehringer, M.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

Feng, W.

Fleddermann, R.

R. Fleddermann, “Interferometry for a space-based gravitational wave observatory reciprocity of an optical fiber,” Ph.D. thesis, Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz UniversitätHannover (2012).

Floberghagen, R.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

García, A.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

García Marin, A. F.

Gerberding, O.

Goßler, S.

Guzmán Cervantes, F.

Haagmans, R.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

Heinzel, G.

Hoyland, D.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Isleif, K.-S.

Jennrich, O.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Johann, U.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Kern, M.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

Kissinger, T.

Köhlenbeck, S.

Kullmann, J.

Lam, T. T.-Y.

McClelland, D. E.

Middleton, K.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Muzi, D.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

Parry, G.

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987).
[Crossref]

Popescu, A.

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

Rabeling, D. S.

Robertson, D.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Rüdiger, A.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Sakai, I.

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987).
[Crossref]

Schilling, R.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Schwarze, T. S.

Shaddock, D.

Shaddock, D. A.

Sheard, B.

Slagmolen, B. J. J.

Sutton, A.

Sutton, A. J.

Tatam, R. P.

Tröbs, M.

Wand, V.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

Wuchenich, D. M.

Youngquist, R.

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987).
[Crossref]

Zheng, J.

J. Zheng, Optical frequency-modulated continuous-wave (FMCW) interferometry (Springer Science & Business Media, 2005), Vol. 107.

Classical Quant. Grav. (1)

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, R. Schilling, U. Johann, and K. Danzmann, “The LTP interferometer and phasemeter,” Classical Quant. Grav. 21, p. 581 (2004)
[Crossref]

J. Lightwave Technol. (1)

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5, 932–940 (1987).
[Crossref]

Opt. Express (8)

T. Kissinger, T. O. Charrett, and R. P. Tatam, “Range-resolved interferometric signal processing using sinusoidal optical frequency modulation,” Opt. Express 23, 9415–9431 (2015).
[Crossref] [PubMed]

O. Gerberding, “Deep frequency modulation interferometry,” Opt. Express 23, 14753–14762 (2015).
[Crossref] [PubMed]

M. Dehne, M. Tröbs, G. Heinzel, and K. Danzmann, “Verification of polarising optics for the LISA optical bench,” Opt. Express 20, 27273–27287 (2012).
[Crossref] [PubMed]

K.-S. Isleif, O. Gerberding, S. Köhlenbeck, A. Sutton, B. Sheard, S. Goßler, D. Shaddock, G. Heinzel, and K. Danzmann, “Highspeed multiplexed heterodyne interferometry,” Opt. Express 22, 24689–24696 (2014).
[Crossref] [PubMed]

A. J. Sutton, O. Gerberding, G. Heinzel, and D. A. Shaddock, “Digitally enhanced homodyne interferometry,” Opt. Express 20, 22195–22207 (2012).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

T. S. Schwarze, O. Gerberding, F. G. Cervantes, G. Heinzel, and K. Danzmann, “Advanced phasemeter for deep phase modulation interferometry,” Opt. Express 22, 18214–18223 (2014).
[Crossref] [PubMed]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

O. Gerberding and et al., “Readout for intersatellite laser interferometry: Measuring low frequency phase fluctuations of high-frequency signals with microradian precision,” Rev. Sci. Instrum. 86, 074501 (2015).
[Crossref] [PubMed]

Other (4)

J. Zheng, Optical frequency-modulated continuous-wave (FMCW) interferometry (Springer Science & Business Media, 2005), Vol. 107.

R. Fleddermann, “Interferometry for a space-based gravitational wave observatory reciprocity of an optical fiber,” Ph.D. thesis, Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz UniversitätHannover (2012).

M. R. Drinkwater, R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, “The GOCE gravity mission: ESAs first core Earth explorer,” in Proceedings of the 3rd international GOCE user workshop, (European Space Agency Noordwijk, The Netherlands, 2006), pp. 6–8.

K. Danzmann and et al., “The Gravitational Universe: Whitepaper for the ESA L2/L3 selection,” (2013).

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Figures (3)

Fig. 1
Fig. 1 Sketch of the experimental set-up. The laser preparation shown in (a) is a fiber-based set-up with amplitude stabilisation. The test bed shown in (b) consists of a Michelson Interferometer (MI) and a Mach-Zehnder Interferometer (MZI) which are free beam setups. Part (c) shows the data post-processing system using an IQ–demodulation and the fit algorithm to extract the phase, amplitude and modulation information.
Fig. 2
Fig. 2 Spectral densities of the phase determined from the frequency domain fit algorithm using the Bessel function amplitudes [5] with the modulation parameters fm = 800Hz, m ≈ 6.16 for the Michelson Interferometer (MI) and m ≈ 9.01 for the Mach-Zehnder Interferometer (MZI). The dark blue line shows the phases φ+ and φ from both interferometric outputs of the MZI. The yellow curve shows the initial phase measurements φa from the MI, the green dashed curve shows the corresponding laser frequency noise corrected phase data φa,corr. The residuals between two measurements which are electronically split are given by φi (red line and light blue line for the MZI and MI). The residuals of the π-combination are given by φπ (purple line). As reference we also plot the typical 1 pm requirement for the displacement sensitivity aimed at in LISA.
Fig. 3
Fig. 3 A typical DFM signal in the time domain of a measured data series for fm = 1kHz (black dots) and a modulation depth of m = 6.47. The function h1(t), given by Eq. (6) and denoting the theoretical DFM signal, fits the data with a sum of squared errors of SSE = 227.1 V2. The function h10(t), given by Eq. (7), includes frequency modulations at harmonics of 1 kHz due to the very deep frequency modulation and has an error of SSE = 8.8 V2. The remaining two curves show the difference of the measured data and the two fit functions.

Equations (7)

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

f DFM ( t ) = Δ f cos ( 2 π f m t + ψ m )
P out ( t ) = E + E κ cos ( φ + 2 π Δ f τ cos [ 2 π f m t + ψ m ] ) ,
m = 2 π Δ f τ ,
φ i , Δ = φ i 1 φ i 2 0 ,
φ π = φ + + φ π .
h 1 ( t ) = E + E κ cos ( φ + m cos [ 2 π f m t + ψ m ] ) .
h 10 ( t ) = E + E κ cos ( φ + k = 1 10 m k cos [ k 2 π f m t + ψ m , k ] ) .

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