Abstract

A white light extrinsic Fabry–Perot interferometer is implemented as a noncontacting displacement sensor, providing robust, absolute displacement measurements with micrometer accuracy at a sampling rate of 10 Hz. This paper presents a dynamic model of the sensing cavity between the sensor probe and the nearby target surface using a Fabry–Perot etalon approach obtained from straightforward electromagnetic field formulations. Such a model is important for system characterization, as the dynamically changing cavity length imparts a Doppler shift on any signals circulating within the sensing cavity. Contrary to previously published results, Doppler-induced shifting within the low-finesse sensing cavity is shown to significantly distort the measurement signal as recorded by the sensor. Experimental and simulation results are compared, and the direct effects of cavity dynamics on the measurement signal are analyzed along with their indirect impact on sensor performance. This document has been approved by Los Alamos National Laboratory for unlimited public release (LA-UR 12-00301).

© 2012 Optical Society of America

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References

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  1. F. Shen and A. Wang, “Frequency-estimation-based signal-processing algorithm for white-light optical fiber Fabry-Perót interferometers,” Appl. Opt. 44, 5206–5214 (2005).
    [CrossRef]
  2. Y. Jiang, “Fourier transform white-light interferometry for the measurement of fiber-optic extrinsic Fabry-Perót interferometric displacement sensors,” IEEE Photon. Technol. Lett. 20, 75–77 (2008).
    [CrossRef]
  3. D. T. Smith, J. R. Pratt, and L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low-frequency displacement measurement,” Rev. Sci. Instr. 80, 035105 (2009).
    [CrossRef]
  4. E. A. Moro, M. D. Todd, and A. D. Puckett, “Using a validated transmission model for the optimization of bundled fiber optic displacement sensors,” Appl. Opt. 50, 6526–6535(2011).
    [CrossRef]
  5. M. Rakhmanov, “Doppler-induced dynamics of fields in Fabry-Perót cavities with suspended mirrors,” Appl. Opt. 40, 1942–1949 (2001).
    [CrossRef]
  6. M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
    [CrossRef]
  7. D. Redding, M. Regehr, and L. Sievers, “Dynamic models of Fabry-Perót interferometers,” Appl. Opt. 41, 2894–2906 (2002).
    [CrossRef]
  8. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry-Perót interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
    [CrossRef]
  9. A. E. Siegman, Lasers (University Science, 1986).
  10. M. Han, “Theoretical and experimental study of low-finesse extrinsic Fabry-Perót interferometric fiber optic sensors,” Ph.D. dissertation (Virginia Tech, 2006).

2011 (1)

2009 (1)

D. T. Smith, J. R. Pratt, and L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low-frequency displacement measurement,” Rev. Sci. Instr. 80, 035105 (2009).
[CrossRef]

2008 (1)

Y. Jiang, “Fourier transform white-light interferometry for the measurement of fiber-optic extrinsic Fabry-Perót interferometric displacement sensors,” IEEE Photon. Technol. Lett. 20, 75–77 (2008).
[CrossRef]

2005 (1)

2002 (2)

D. Redding, M. Regehr, and L. Sievers, “Dynamic models of Fabry-Perót interferometers,” Appl. Opt. 41, 2894–2906 (2002).
[CrossRef]

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

2001 (1)

1999 (1)

Byer, R. L.

Gustafson, E. K.

Han, M.

M. Han, “Theoretical and experimental study of low-finesse extrinsic Fabry-Perót interferometric fiber optic sensors,” Ph.D. dissertation (Virginia Tech, 2006).

Howard, L. P.

D. T. Smith, J. R. Pratt, and L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low-frequency displacement measurement,” Rev. Sci. Instr. 80, 035105 (2009).
[CrossRef]

Husman, M. E.

Jiang, Y.

Y. Jiang, “Fourier transform white-light interferometry for the measurement of fiber-optic extrinsic Fabry-Perót interferometric displacement sensors,” IEEE Photon. Technol. Lett. 20, 75–77 (2008).
[CrossRef]

Lawrence, M. J.

Moro, E. A.

Pratt, J. R.

D. T. Smith, J. R. Pratt, and L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low-frequency displacement measurement,” Rev. Sci. Instr. 80, 035105 (2009).
[CrossRef]

Puckett, A. D.

Rakhmanov, M.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

M. Rakhmanov, “Doppler-induced dynamics of fields in Fabry-Perót cavities with suspended mirrors,” Appl. Opt. 40, 1942–1949 (2001).
[CrossRef]

Redding, D.

Regehr, M.

Reitze, D. H.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Savage, R. L.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Shen, F.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Sievers, L.

Smith, D. T.

D. T. Smith, J. R. Pratt, and L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low-frequency displacement measurement,” Rev. Sci. Instr. 80, 035105 (2009).
[CrossRef]

Tanner, D. B.

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Todd, M. D.

Wang, A.

Willke, B.

Appl. Opt. (4)

IEEE Photon. Technol. Lett. (1)

Y. Jiang, “Fourier transform white-light interferometry for the measurement of fiber-optic extrinsic Fabry-Perót interferometric displacement sensors,” IEEE Photon. Technol. Lett. 20, 75–77 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Lett. A (1)

M. Rakhmanov, R. L. Savage, D. H. Reitze, and D. B. Tanner, “Dynamic resonance of light in Fabry-Perót cavities,” Phys. Lett. A 305, 239–244 (2002).
[CrossRef]

Rev. Sci. Instr. (1)

D. T. Smith, J. R. Pratt, and L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low-frequency displacement measurement,” Rev. Sci. Instr. 80, 035105 (2009).
[CrossRef]

Other (2)

A. E. Siegman, Lasers (University Science, 1986).

M. Han, “Theoretical and experimental study of low-finesse extrinsic Fabry-Perót interferometric fiber optic sensors,” Ph.D. dissertation (Virginia Tech, 2006).

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Figures (10)

Fig. 1.
Fig. 1.

General schematic is shown for one kind of swept-filter, white light EFPI displacement sensor.

Fig. 2.
Fig. 2.

This diagram illustrates the relationships between the different fields present in a Fabry–Perot cavity, which is bounded by surface 1 and surface 2. The reflected field Erefl is a combination of a partial reflection of the incident field Einc and a partial transmission of the circulating field Ecirc, and the transmitted field Etrans is a partial transmission of the circulating field Ecirc.

Fig. 3.
Fig. 3.

Simulated filtered signal has a single peak in the intensity versus wavelength (or frequency) domain.

Fig. 4.
Fig. 4.

Simulated measurement seen by the photodetector is shown here.

Fig. 5.
Fig. 5.

Fourier analysis of photodetector’s simulated, measured signal (Fig. 4) reveals the cavity’s FSR, which may be related back to the cavity length. The peak at 5 mm corresponds to the cavity’s true length, while the peak at 10 mm is a harmonic of the primary peak.

Fig. 6.
Fig. 6.

Experimental setup is shown here. The optical source, swept filter, circulator, referencing hardware, photodetectors, and data logging hardware are all included within the optical interrogator and its chassis.

Fig. 7.
Fig. 7.

Experimental and simulated spectra are shown for the static case, with the simulation results indicated by dashed curves. The locations and magnitudes of the simulated and experimental peaks are in agreement.

Fig. 8.
Fig. 8.

Experimental and simulated spectra are shown for the dynamic case. The locations of the simulated and experimental peaks are in agreement. The Doppler induced shift results in a cavity length bias of approximately 0.65*v (in mm), where v is the DUT velocity in mm/s.

Fig. 9.
Fig. 9.

Doppler shifted peak location data are shown (a) for L=56mm with a velocity of 0.1mm/s and (b)–(d) for L=1020mm with various velocities over the range 110mm/s. The anomaly at t=10.6s in (d) occurred when the peak detection algorithm lost track of the true, cavity length peak.

Fig. 10.
Fig. 10.

Experimental peak location data are shown for (a) L=56mm and (b) L=2425mm , where in both cases the position was changing at a rate of 0.1mm/s. The Doppler induced bias may be seen when the linear stage stops moving (about 25 s) and starts moving again (about 27 s).

Tables (1)

Tables Icon

Table 1. Physical Parameters Assumed during This Analysis

Equations (11)

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Ecirc(t)=jt1Einc(t)+grt(t)Ecirc(tτ),
grt(t(n1)τ)=r1r2exp[2α0L(tnτ/2)j2ω(tnτ)L(tnτ/2)c].
Ecirc(t)=jt1{grt(t)Einc,D(tτ)+grt(t)grt(tτ)Einc,D(t2τ)++grt(t)grt(tτ)grt(t(N1)τ)Einc,D(tNτ)}.
Einc,D(tnτ)=E0exp[jD(tnτ/2)(k(tnτ)xω(tnτ)(tnτ))],
D(tnτ/2)=1+2L(tnτ/2)c,
Erefl(t)=r1Einc(t)+jt1r1Ecirc(t),
Erefl(t)=r1E0exp[jω(t)t]t12r1i=110τS/τ[grt(t(i1)τ)iE0exp[jDi(tiτ/2)ω(tiτ)(tiτ)]].
Irefl(t)=12cε0EreflErefl*,
τS=τln(r1r2exp(2α0L)).
I(t)=tf41+rf42rfcos(2ω(t)Lf(t)/c).
Finesse=πr1r21r1r2,

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