Abstract

We propose and demonstrate sub-micron displacement sensing and sensitivity enhancement using a two-frequency interferometer and a Kerr phase-interrogator. Displacement induces phase variation on a sinusoidally modulated optical signal by changing the length of the path that either of the signal’s two spectral components propagates through. A Kerr phase-interrogator converts the resulting phase variation into power variation allowing for sub-micron displacement sensing. The sensitivity of this novel displacement sensor is enhanced beyond the wavelength-limited sensitivity of the widely used Michelson interferometric displacement sensor. The proposed approach for sensitivity enhancement creates a whole new class of sensors with ultra-high sensitivity.

© 2015 Optical Society of America

1. Introduction

Sub-micron interferometric displacement sensors (IDS) based on homodyne interferometers have been widely used as surface profile measurement devices, velocity meters, and all-optical vibration transducers [1–3]. The most common homodyne interferometric displacement sensor is implemented using a Michelson interferometer. The Michelson interferometric displacement sensor converts displacement-induced variation on the phase of a laser into power variation allowing for sub-micron displacement measurement. The power variation with displacement in a Michelson IDS is given by P(D) = Pmax cos2 (2πD/λ +ϕ0) where λ is the laser wavelength, D is the displacement, and ϕ0 is the initial phase at D = 0 [4]. The sensitivity (σ) of a Michelson IDS, defined as the maximum variation-rate of P/Pmax with D, is given by σ = 2π/λ which is limited by the wavelength λ of the laser.

Heterodyne interferometers, such as the two-frequency IDS [5], have also been widely used for sub-micron displacement measurement. A two-frequency IDS converts displacement-induced variation on the phase of a laser into a corresponding variation of the phase of a sinusoidally-modulated optical signal (SMOS). The power of the SMOS is given by P(t,D) = Pmax cos2 [π fst + 2πD/λ − ϕ0] where fs is the power modulation frequency [5]. Existing implementations of two-frequency IDS convert the SMOS into a sinusoidal electrical signal using a photo-detector and then retrieve the displacement value using analog or digital phase demodulation techniques. Utilization of advanced phase demodulation techniques in a two-frequency IDS opens the way for sensitivity enhancement beyond the wavelength limit that is inherent in a Michelson IDS.

Recently, we have demonstrated a novel non-interferometric displacement sensor with micron-level resolution based on a Kerr phase-interrogator [6]. In this approach, displacement induces variation on the phase of a SMOS by changing the length of the path through which the SMOS propagates. Then, a Kerr phase-interrogator converts the resulting phase variation into power variation using all-optical signal processing by the nonlinear Kerr effect [6]. The power as a function of displacement is given by P(D) = Pmax cos2 (2πD/λs +ϕ0), where λs = vg/fs is the wavelength of the SMOS with a typical value of several millimeters, and vg is the group velocity. The sensitivity of this displacement sensor is given by σ = 2π/λs which is much lower than that of a Michelson IDS because λs is typically three orders of magnitude longer than the wavelength λ of the laser of the Michelson IDS.

Sensitivity higher than that of a Michelson IDS can be achieved by utilization of a hybrid system composed of a Kerr phase-interrogator and a two-frequency IDS. In this hybrid displacement sensor (HDS), a two-frequency IDS converts the displacement-induced laser phase variation into SMOS phase variation using optical interference, and then, a Kerr phase-interrogator converts the SMOS phase variation into power variation using the nonlinear Kerr effect. Similar to a Michelson IDS, the HDS all-optically converts laser phase variation into power variation; however, the HDS has higher sensitivity than a Michelson IDS because the advanced demodulation scheme of the Kerr phase-interrogator eliminates the wavelength limitation on sensitivity.

In this paper, we propose and demonstrate a HDS for sub-micron displacement measurement and sensitivity enhancement beyond the wavelength limit of a Michelson IDS. A novel experimental setup combines a Kerr phase-interrogator and a two-frequency IDS to embody the proposed HDS. Theoretical analysis shows that displacement induces phase variation on one of the spectral components of a SMOS leading to a corresponding phase variation on the SMOS. A Kerr phase-interrogator is utilized to convert phase-variation of the SMOS into power-variation allowing for sub-micron displacement measurement. Investigation of the Kerr phase-interrogator when the peak of the nonlinear Kerr-induced phase modulation ϕSPM is greater than 10 radians reveals that the sensitivity of a HDS is σ = 0.36ϕSPM × 2π/λ, which is higher than the sensitivity of a Michelson IDS by a factor of 0.36ϕSPM. Finally, experimental results demonstrate displacement sensing at sub-micron resolution and sensitivity enhancement in agreement with the theoretical model.

2. Experimental setup

Figure 1(a) presents a schematic of the sub-wavelength displacement measurement setup comprised of a Kerr phase-interrogator and a two-frequency IDS. A continuous-wave (CW) laser (RIO) operating at a wavelength λ0 is amplitude-modulated using a sinusoidal electrical signal generator (HP 83752A) oscillating at fm = 10 GHz to obtain a SMOS with power oscillating at a frequency fs = 2fm. The optical spectrum of the SMOS is composed of two distinct peaks at λ1 and λ2 separated by Δλ=λ02fs/c with c being the speed of light in vacuum, as illustrated in Fig. 1. A fiber-coupled polarization beam splitter (from General Photonics) divides the power of the SMOS into a sensor path and a reference path.

 figure: Fig. 1

Fig. 1 Schematic of the sub-micron displacement measurement setup based on a Kerr phase-interrogator and a two-frequency IDS. RF: radio-frequency; CW: continuous-wave; EOM: electro-optic modulator; PSD: power spectral density; PC: polarization controller; FPS: fiber polarization splitter; FPC: fiber polarization combiner; PM: polarization maintaining; PBS: polarization beam splitter; M: mirror; and EDFA: Erbium-doped fiber amplifier.

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The SMOS in the sensor path propagates through a polarization-maintaining circulator (from General Photonics) and is launched with linear polarization at 45° from the principal axis of a 10 m long polarization-maintaining fiber with a beat-length of 2 mm. Due to the wavelength dependence of birefringence in the polarization-maintaining fiber, the two wavelength components of the SMOS exit the birefringent fiber at port (b) with orthogonal polarizations [7]. A free-space polarization-beam-splitter separates the two orthogonally polarized spectral components, as illustrated in Fig. 1. Each spectral components Ei at λi travels a separate path with length Li and is reflected by a mirror Mi. Mirror M1 is attached to a piezo-electric actuator (Thorlabs PK2FQP2) and displacement of M1 is induced by variation of the voltage applied to the piezo-electric actuator from a voltage source (HP E3631A). The displacement of M1 leads to variation in the phase of E1 given by ϕ1 = k1L1 +ϕ1,0 = 2k1D +ϕ1,0, where k1 = 2π/λ1, D is the displacement distance, and ϕ1,0 is a constant. Mirror M2 is fixed and the phase of E2 is given by ϕ2 = ϕ2,0, where ϕ2,0 is a constant.

After reflection from mirrors Mi, the spectral components travel back through the polarization-maintaining fiber and exit with parallel polarizations at port (a) to reconstruct the SMOS. The SMOS travels back through the polarization-maintaining circulator and a fiber-coupled polarization beam combiner (from General Photonics) recombines the signals from the sensor and the reference paths. The combined signal at the output of the fiber polarization combiner is amplified using an Erbium-doped fiber amplifier (Amonics AEDFA-33-B-FA) and is launched into a nonlinear Kerr medium comprised of a fiber with a length of Lkerr = 5.7 km, a loss coefficient of αdB = 0.47 dB/km measured using the cut-back method, a coupling-loss of 2 dB at each fiber end, a waveguide nonlinearity of γ = 4.2 W−1km−1 measured as described in [8], and a chromatic-dispersion of Dc = 3 ps/nm-km measured using the modulation phase-shift method [9, 10]. The electric field amplitudes of the SMOSs from the sensor and the reference paths are given by

A=Pp/2cos(πfst+ϕ),
A=Pp/2cos(πfst+ϕ),
respectively, where ϕ = k1D + (ϕ1,0 −ϕ2,0)/2, ϕ = ϕref with ϕref being a constant, D is the displacement of M1, and Pp/2 is the peak power of each SMOS. Self-induced phase-modulation in the Kerr medium leads to the formation of distinct sidebands Pi with i = 1,2,3,… [6, 8, 10–12]. The power of the first-order side-band P1 as a function of D when the maximum nonlinear phase-shift accumulated in the Kerr medium ϕSPM = γPpLeff with Leff being the effective-length of the Kerr medium Leff = [1 exp(−αLkerr)] satisfies the condition ϕSPM < 0.5 is given by [6]
P1=P1maxcos2(ϕ)
where ϕ = ϕ − ϕ = 2πD/λ1 + ϕ0 with ϕ0 = (ϕ1,0 −ϕ2,0)/2 − ϕref, and the displacement measurement sensitivity is
σ=max{|dP1/dD|/P1max}=2π/λ1.

3. Sensitivity enhancement

Sensitivity enhancement is achieved when the Kerr phase-interrogator is operated under the condition ϕSPM 0.5 where the power of the first-order side-band is given by [6]

P1(ϕ)=Pp×{J12[(ϕSPM/2)cos(ϕ)]+J22[(ϕSPM/2)cos(ϕ)]}
where Jn (x) is the Bessel function of the first kind. Figure 2 presents the calculated value of P1 as a function of ϕ for ϕSPM = 0.5, 5, 10, and 50 illustrating that the variation of P1 with ϕ becomes faster around ϕ = (0.5 + m)π with m being an integer. The value of P1 as a function of ϕ around ϕ = (0.5 + m)π is approximated by
P1(ϕ)=P1maxsin2[0.36×ϕSPM×(ϕπ/2mπ)]
for all values of ϕ that satisfy −π/2 < 0.36 ×ϕSPM × (ϕ−π/2−mπ) < π/2. Figure 3 shows that Eq. (6) is valid for ϕSPM > 10 and has been verified for ϕSPM as large as 104 and beyond. Using Eq. (6), the displacement measurement sensitivity is
σ=max{|dP1/dD|/P1max}=0.36ϕSPM×(2π/λ1),
which by comparison with the sensitivity under the condition ϕSPM < 0.5 in Eq. (4) indicates an enhancement by a factor of 0.36ϕSPM.

 figure: Fig. 2

Fig. 2 Normalized power of the first order-sideband as a function of phase calculated using Eq. (5) for ϕSPM = 0.5, 5, 10, and 50.

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 figure: Fig. 3

Fig. 3 Exact and approximate values of P1 as a function of ϕ calculated using Eq. (5) and Eq. (6) with a) ϕSPM = 5, b) ϕSPM = 10, and c) ϕSPM = 1000.

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4. Experimental results and discussion

To demonstrate displacement measurement under the condition ϕSPM < 0.5, the combined signal at the output of the polarization beam combiner is amplified to 40 mW and then launched into the Kerr medium. The power of the first order side-band is recorded as the voltage of the piezo-electric actuator is increased in steps of 0.1 V corresponding to a displacement step of 28.5 nm. Figure 4 presents the measured values of P1 as a function of displacement showing sinusoidal dependence of the side-band power with displacement as predicted by Eq. (3). Also presented in Fig. 4 is the theoretical value of P1 (D) obtained from Eq. (3) with λ1 = 1548 nm showing close agreement between theory and experiment. The measured sensitivity from Fig. 4 is σ = 3.86×106 m−1 in close agreement with the the theoretically calculated value σ = 4.06 × 106 m−1 obtained from Eq. (4) with λ1 = 1548 nm.

 figure: Fig. 4

Fig. 4 Experimentally measured and theoretically calculated side-band power as a function of displacement for ϕSPM < 0.5.

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To demonstrate sensitivity enhancement, the power at the input of the Kerr medium is further amplified to 450 mW corresponding to ϕSPM = 5 and the value of P1 is measured as the mirror M1 is displaced in steps of 28.5 nm. Figure 5 presents the measured values of P1 as a function of displacement along with the theoretical value calculated using Eq. (5) with λ1 = 1548 nm and ϕSPM = 5 showing close agreement between theory and experiment. A sensitivity of σ = 6.76 × 106 m−1 is measured from Fig. 5 indicating a sensitivity enhancement by a factor of 1.75 in comparison with the sensitivity at ϕSPM < 0.5. Figure 3(a) shows that the approximate value of P1 from Eq. (6) for ϕ in the range determined by the condition −π/4 < 0.36×ϕSPM ×(ϕ −π/2 − mπ) < π/4 correponding to P1/P1max < 0.5 closely matches the exact value of P1 from Eq. (5); therefore, Eq. (7) provides a close estimate of the sensitivity at ϕSPM = 5. Using Eq. (7), the theoretical estimate of the sensitivity enhancement is 0.36ϕSPM = 1.8 in close agreement with the experimentally measured value.

 figure: Fig. 5

Fig. 5 Experimental measurements of power as a function of displacement for ϕSPM = 5.

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Optimal displacement sensing is achieved around operating points where the power variation with displacement is maximum. For ϕSPM < 0.5, the operating points are located at the quadrature points around Dop such that ϕ = 2πDop1 + (ϕ1,0 −ϕ2,0)/2−ϕref = π/4 + qπ/2 with q being an integer. For ϕSPM > 10, the operating points from Eq. (6) are located at Dop where ϕ satisfies the condition 0.36 ×ϕSPM × (ϕ−π/2−mπ) = ±π/4. The operating point can be shifted by using a variable delay-line in the reference arm to vary t and change ϕref = π fst [6].

The displacement resolution under the condition ϕSPM < 0.5 is calculated from the minimum detectable phase-change Δϕ = 2πΔD/λ1 = α leading to ΔD = αλ1/2π. Differentiation of Eq. (3) around the operating points leads to |δϕ|=|δP1/P1max|, where δϕ represents the fluctuations of phase-shift and δP1 represent the noise-induced power fluctuations. With a maximum power fluctuation of 1%, the minimum resolvable differential phase-shift is α = max {|δϕ|} = 10−2 and the displacement resolution is ΔD = 2.46 nm for λ1 = 1548 nm. Under the condition ϕSPM > 10, the displacement resolution is calculated from 0.36ϕSPM × Δϕ = 0.36ϕSPM ×2πΔD/λ1 =α which leads to ΔD =αλ1/(0.72πϕSPM) indicating that the displacement resolution is refined by a factor of 0.36ϕSPM.

The dynamic-range DDR is defined as the quasi-linear range over which P1 varies from 0.2Pmax to 0.8Pmax around the operating points to obtain a one-to-one correspondence between P1 and D [13]. For ϕSPM < 0.5, P1 = 0.2Pmax at 2πDmin1 + ϕ0 = 0.35π and P1 = 0.8Pmax at 2πDmax1 +ϕ0 = 0.15π leading to a dynamic range DDR = |Dmin − Dmax| = 0.1λ1. For λ1 = 1548 nm, the dynamic-range is DDR = 154.8nm and the number of resolvable points within this range is the rounded ratio between the dynamic-range and the displacement resolution ⌊DDRD⌋ = ⌊0.2π/α⌋ = 62. Similarly, for ϕSPM > 10, P1 = 0.2Pmax at 0.36 × ϕSPM × (2πDmin1 +ϕ0 −π/2 − mπ) = 0.15π and P1 = 0.8Pmax at 0.36 × ϕSPM × (2πDmax1 +ϕ0 −π/2 − mπ) = 0.35π leading to a dynamic-range DDR = 0.1λ1/(0.36ϕSPM) which corresponds to a reduction by a factor of 0.36ϕSPM when compared with DDR for ϕSPM < 0.5. The number of resolvable points when ϕSPM > 10 is given by ⌊DDRD⌋ = ⌊0.2π/α⌋ which is identical to the number of resolvable points when ϕSPM < 0.5.

5. Comparison with Michelson interferometric displacement sensors

The power as a function of displacement for a Michelson IDS is given by P(ϕ) = Pmax cos2 (ϕ) with ϕ = 2πD/λ +ϕ0 [4] which is identical to Eq. (3). Therefore, the sensitivity of a Michelson IDS is the same as the sensitivity of a HDS when the Kerr phase-interrogator operates under the condition ϕSPM < 0.5. However, when the Kerr phase-interrogator operates under the condition ϕSPM > 10, the power as a function of displacement for the HDS is given by Eq. (6). In this case, the sensitivity of the HDS given by Eq. (7) is higher than that of the Michelson IDS by a factor 0.36ϕSPM indicating sensitivity enhancement beyond the wavelength limit of a Michelson IDS. This sensitivity enhancement can be adapted for a variety of other sensing applications that utilize Michelson interferometers such as vibration monitoring, gravitational wave detection, temperature/strain measurement, and refractive-index sensing.

6. Conclusion

A novel hybrid displacement sensor for sub-micron displacement measurement is obtained by combining a Kerr phase-interrogator and a two-frequency interferometric displacement sensor. The advanced demodulation scheme of the Kerr phase-interrogator enhances the sensitivity of the hybrid displacement sensor beyond the wavelength limited sensitivity of a Michelson interferometric displacement sensor. A sub-micron displacement sensor with a sensitivity enhancement factor of 1.75 is experimentally demonstrated. Future work will focus on utilizing this sensitivity enhancement approach in novel devices for vibration monitoring and refractive-index sensing.

Acknowledgments

The authors are thankful to the NSERC Discovery Grant and Canada Research Chair Program (CRC in Fiber Optics and Photonics).

References and links

1. F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48, 603–609 (1970). [CrossRef]  

2. C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972). [CrossRef]  

3. B. J. Pernick, “Self-consistent and direct reading laser homodyne measurement technique,” Appl. Opt. 12, 607–610 (1973). [CrossRef]   [PubMed]  

4. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991). [CrossRef]  

5. S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990). [CrossRef]  

6. C. Baker and X. Bao, “Displacement sensor based on kerr induced phase-modulation of orthogonally polarized sinusoidal optical signals,” Opt. Express 22, 9095–9100 (2014). [CrossRef]   [PubMed]  

7. S. C. Rashleigh, “Wavelength dependence of birefringence in highly birefringent fibers,” Opt. Lett. 7, 294–296 (1982). [CrossRef]   [PubMed]  

8. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 μ m,” Opt. Lett. 21, 1966–1968 (1996). [CrossRef]   [PubMed]  

9. B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982). [CrossRef]  

10. C. Baker, Y. Lu, and X. Bao, “Chromatic-dispersion measurement by modulation phase-shift method using a Kerr phase-interrogator,” Opt. Express 22, 22314–22319 (2014). [CrossRef]   [PubMed]  

11. M. Rochette, C. Baker, and R. Ahmad, “All-optical polarization-mode dispersion monitor for return-to-zero optical signals at 40 gbits/s and beyond,” Opt. Lett. 35, 3703–3705 (2010). [CrossRef]   [PubMed]  

12. C. Baker, Y. Lu, J. Song, and X. Bao, “Incoherent optical frequency domain reflectometry based on a kerr phase-interrogator,” Opt. Express 22, 15370–15375 (2014). [CrossRef]   [PubMed]  

13. Y. Lu, C. Baker, L. Chen, and X. Bao, “Group-delay based temperature sensing in linearly-chirped fiber Bragg gratings using a Kerr phase-interrogator,” J. Lightwave Technol. 33, 381-385 (2014).

References

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  1. F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48, 603–609 (1970).
    [Crossref]
  2. C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972).
    [Crossref]
  3. B. J. Pernick, “Self-consistent and direct reading laser homodyne measurement technique,” Appl. Opt. 12, 607–610 (1973).
    [Crossref] [PubMed]
  4. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [Crossref]
  5. S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990).
    [Crossref]
  6. C. Baker and X. Bao, “Displacement sensor based on kerr induced phase-modulation of orthogonally polarized sinusoidal optical signals,” Opt. Express 22, 9095–9100 (2014).
    [Crossref] [PubMed]
  7. S. C. Rashleigh, “Wavelength dependence of birefringence in highly birefringent fibers,” Opt. Lett. 7, 294–296 (1982).
    [Crossref] [PubMed]
  8. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 μ m,” Opt. Lett. 21, 1966–1968 (1996).
    [Crossref] [PubMed]
  9. B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
    [Crossref]
  10. C. Baker, Y. Lu, and X. Bao, “Chromatic-dispersion measurement by modulation phase-shift method using a Kerr phase-interrogator,” Opt. Express 22, 22314–22319 (2014).
    [Crossref] [PubMed]
  11. M. Rochette, C. Baker, and R. Ahmad, “All-optical polarization-mode dispersion monitor for return-to-zero optical signals at 40 gbits/s and beyond,” Opt. Lett. 35, 3703–3705 (2010).
    [Crossref] [PubMed]
  12. C. Baker, Y. Lu, J. Song, and X. Bao, “Incoherent optical frequency domain reflectometry based on a kerr phase-interrogator,” Opt. Express 22, 15370–15375 (2014).
    [Crossref] [PubMed]
  13. Y. Lu, C. Baker, L. Chen, and X. Bao, “Group-delay based temperature sensing in linearly-chirped fiber Bragg gratings using a Kerr phase-interrogator,” J. Lightwave Technol. 33, 381-385 (2014).

2014 (4)

2010 (1)

1996 (1)

1990 (1)

S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990).
[Crossref]

1982 (2)

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[Crossref]

S. C. Rashleigh, “Wavelength dependence of birefringence in highly birefringent fibers,” Opt. Lett. 7, 294–296 (1982).
[Crossref] [PubMed]

1973 (1)

1972 (1)

C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972).
[Crossref]

1970 (1)

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48, 603–609 (1970).
[Crossref]

Ahmad, R.

Andrews, F. A.

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48, 603–609 (1970).
[Crossref]

Baker, C.

Bao, X.

Bartlett, S.

S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990).
[Crossref]

Boskovic, A.

Brown, C. R.

C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972).
[Crossref]

Brown, G. R.

C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972).
[Crossref]

Chen, L.

Chernikov, S. V.

Costa, B.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[Crossref]

Eberhardt, F. J.

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48, 603–609 (1970).
[Crossref]

Farahi, F.

S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990).
[Crossref]

Gruner-Nielsen, L.

Jackson, D.

S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990).
[Crossref]

Levring, O. A.

Lu, Y.

Mazzoni, D.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[Crossref]

Niblett, D. H.

C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972).
[Crossref]

Pernick, B. J.

Puleo, M.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[Crossref]

Rashleigh, S. C.

Rochette, M.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

Song, J.

Taylor, J. R.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

Vezzoni, E.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using led’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[Crossref]

J. Acoust. Soc. Am. (1)

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48, 603–609 (1970).
[Crossref]

J. Lightwave Technol. (1)

J. Phys. E (1)

C. R. Brown, G. R. Brown, and D. H. Niblett, “Measurement of small strain amplitudes in internal friction experiments by means of a laser interferometer,” J. Phys. E 5, 966 (1972).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Rev. Sci. Instrum. (1)

S. Bartlett, F. Farahi, and D. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014–1017 (1990).
[Crossref]

Other (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic of the sub-micron displacement measurement setup based on a Kerr phase-interrogator and a two-frequency IDS. RF: radio-frequency; CW: continuous-wave; EOM: electro-optic modulator; PSD: power spectral density; PC: polarization controller; FPS: fiber polarization splitter; FPC: fiber polarization combiner; PM: polarization maintaining; PBS: polarization beam splitter; M: mirror; and EDFA: Erbium-doped fiber amplifier.
Fig. 2
Fig. 2 Normalized power of the first order-sideband as a function of phase calculated using Eq. (5) for ϕSPM = 0.5, 5, 10, and 50.
Fig. 3
Fig. 3 Exact and approximate values of P1 as a function of ϕ calculated using Eq. (5) and Eq. (6) with a) ϕSPM = 5, b) ϕSPM = 10, and c) ϕSPM = 1000.
Fig. 4
Fig. 4 Experimentally measured and theoretically calculated side-band power as a function of displacement for ϕSPM < 0.5.
Fig. 5
Fig. 5 Experimental measurements of power as a function of displacement for ϕSPM = 5.

Equations (7)

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A = P p / 2 cos ( π f s t + ϕ ) ,
A = P p / 2 cos ( π f s t + ϕ ) ,
P 1 = P 1 max cos 2 ( ϕ )
σ = max { | d P 1 / d D | / P 1 max } = 2 π / λ 1 .
P 1 ( ϕ ) = P p × { J 1 2 [ ( ϕ S P M / 2 ) cos ( ϕ ) ] + J 2 2 [ ( ϕ S P M / 2 ) cos ( ϕ ) ] }
P 1 ( ϕ ) = P 1 max sin 2 [ 0.36 × ϕ S P M × ( ϕ π / 2 m π ) ]
σ = max { | d P 1 / d D | / P 1 max } = 0.36 ϕ S P M × ( 2 π / λ 1 ) ,

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