Abstract

The influence of reflector losses attracts little discussion in standard treatments of the Fabry–Perot interferometer yet may be an important factor contributing to errors in phase-stepped demodulation of fiber optic Fabry–Perot (FFP) sensors. We describe a general transfer function for FFP sensors with complex reflection coefficients and estimate systematic phase errors that arise when the asymmetry of the reflected fringe system is neglected, as is common in the literature. The measured asymmetric response of higher-finesse metal–dielectric FFP constructions corroborates a model that predicts systematic phase errors of 0.06 rad in three-step demodulation of a low-finesse FFP sensor (R = 0.05) with internal reflector losses of 25%.

© 2000 Optical Society of America

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References

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    [CrossRef] [PubMed]
  3. H. Singh, J. S. Sirkis, “Simultaneously measuring temperature and strain using optical fiber microcavities,” J. Lightwave Technol. 15, 647–653 (1997).
    [CrossRef]
  4. B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1997 (2)

H. Singh, J. S. Sirkis, “Simultaneously measuring temperature and strain using optical fiber microcavities,” J. Lightwave Technol. 15, 647–653 (1997).
[CrossRef]

W. N. MacPherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps,” Proc. Inst. Electr. Eng. Optoelectron. 144(3) , 130–133 (1997).
[CrossRef]

1996 (1)

1995 (2)

A. Ezbiri, R. P. Tatam, “Passive signal processing for a miniature Fabry–Perot interferometric sensor with a multimode laser-diode source,” Opt. Lett. 20, 1818–1820 (1995).
[CrossRef]

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

1992 (1)

M. N. Inci, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Fabrication of single-mode fibre optic Fabry–Perot interferometers using fusion spliced titanium dioxide optical coatings,” J. Meas. Sci. Technol. 3, 678–684 (1992).
[CrossRef]

1991 (1)

1989 (1)

1988 (2)

C. E. Lee, H. F. Taylor, “Interferometric optical fiber sensors using internal mirrors,” Electron. Lett. 24, 193–194 (1988).
[CrossRef]

C. E. Lee, R. A. Atkins, H. F. Taylor, “Performance of a fiber-optic temperature sensor from -200 °C to 1050 °C,” Opt. Lett. 13, 1038–1040 (1988).
[CrossRef] [PubMed]

1987 (1)

1969 (1)

P. G. Kard, “Theory of Optical Coatings,” Izv. Akad. Nauk Est. SSR, 12, 359 (1969)[reviewed by Z. Knittl in Optics of Thin Films (Wiley, New York, 1976)].

1957 (1)

1949 (1)

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. 62, 405–417 (1949).
[CrossRef]

1906 (1)

M. Hamy, “Fringes by reflection from silvered glass,” J. Phys. Radium 5, 789–809 (1906).

Atkins, R. A.

Barton, J. S.

W. N. MacPherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps,” Proc. Inst. Electr. Eng. Optoelectron. 144(3) , 130–133 (1997).
[CrossRef]

M. N. Inci, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Fabrication of single-mode fibre optic Fabry–Perot interferometers using fusion spliced titanium dioxide optical coatings,” J. Meas. Sci. Technol. 3, 678–684 (1992).
[CrossRef]

Bennet, J. M.

Bernabeu, E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 13, pp. 629–633.

Cownie, A. R.

Ezbiri, A.

Hamy, M.

M. Hamy, “Fringes by reflection from silvered glass,” J. Phys. Radium 5, 789–809 (1906).

Hariharan, P.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4, p. 58.

Holden, J.

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. 62, 405–417 (1949).
[CrossRef]

Inci, M. N.

M. N. Inci, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Fabrication of single-mode fibre optic Fabry–Perot interferometers using fusion spliced titanium dioxide optical coatings,” J. Meas. Sci. Technol. 3, 678–684 (1992).
[CrossRef]

Jones, J. D. C.

W. N. MacPherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps,” Proc. Inst. Electr. Eng. Optoelectron. 144(3) , 130–133 (1997).
[CrossRef]

M. N. Inci, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Fabrication of single-mode fibre optic Fabry–Perot interferometers using fusion spliced titanium dioxide optical coatings,” J. Meas. Sci. Technol. 3, 678–684 (1992).
[CrossRef]

Kao, T. W.

Kard, P. G.

P. G. Kard, “Theory of Optical Coatings,” Izv. Akad. Nauk Est. SSR, 12, 359 (1969)[reviewed by Z. Knittl in Optics of Thin Films (Wiley, New York, 1976)].

Kidd, S. R.

W. N. MacPherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps,” Proc. Inst. Electr. Eng. Optoelectron. 144(3) , 130–133 (1997).
[CrossRef]

M. N. Inci, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Fabrication of single-mode fibre optic Fabry–Perot interferometers using fusion spliced titanium dioxide optical coatings,” J. Meas. Sci. Technol. 3, 678–684 (1992).
[CrossRef]

Lee, C. E.

C. E. Lee, H. F. Taylor, “Interferometric optical fiber sensors using internal mirrors,” Electron. Lett. 24, 193–194 (1988).
[CrossRef]

C. E. Lee, R. A. Atkins, H. F. Taylor, “Performance of a fiber-optic temperature sensor from -200 °C to 1050 °C,” Opt. Lett. 13, 1038–1040 (1988).
[CrossRef] [PubMed]

MacLeod, H. A.

H. A. MacLeod, Thin Film Optical Filters, 2nd ed. (Hilger/McGraw-Hill, New York, 1986).
[CrossRef]

MacPherson, W. N.

W. N. MacPherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps,” Proc. Inst. Electr. Eng. Optoelectron. 144(3) , 130–133 (1997).
[CrossRef]

Monzón, J. J.

Sánchez-Soto, L. L.

Singh, H.

H. Singh, J. S. Sirkis, “Simultaneously measuring temperature and strain using optical fiber microcavities,” J. Lightwave Technol. 15, 647–653 (1997).
[CrossRef]

Sirkis, J. S.

H. Singh, J. S. Sirkis, “Simultaneously measuring temperature and strain using optical fiber microcavities,” J. Lightwave Technol. 15, 647–653 (1997).
[CrossRef]

Surrel, Y.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Tatam, R. P.

Taylor, H. F.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 13, pp. 629–633.

Zhao, B.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (1)

C. E. Lee, H. F. Taylor, “Interferometric optical fiber sensors using internal mirrors,” Electron. Lett. 24, 193–194 (1988).
[CrossRef]

Izv. Akad. Nauk Est. SSR (1)

P. G. Kard, “Theory of Optical Coatings,” Izv. Akad. Nauk Est. SSR, 12, 359 (1969)[reviewed by Z. Knittl in Optics of Thin Films (Wiley, New York, 1976)].

J. Lightwave Technol. (1)

H. Singh, J. S. Sirkis, “Simultaneously measuring temperature and strain using optical fiber microcavities,” J. Lightwave Technol. 15, 647–653 (1997).
[CrossRef]

J. Meas. Sci. Technol. (1)

M. N. Inci, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Fabrication of single-mode fibre optic Fabry–Perot interferometers using fusion spliced titanium dioxide optical coatings,” J. Meas. Sci. Technol. 3, 678–684 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. Radium (1)

M. Hamy, “Fringes by reflection from silvered glass,” J. Phys. Radium 5, 789–809 (1906).

Opt. Eng. (1)

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Opt. Lett. (3)

Proc. Inst. Electr. Eng. Optoelectron. (1)

W. N. MacPherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fibre Fabry–Perot sensors with inexact phase steps,” Proc. Inst. Electr. Eng. Optoelectron. 144(3) , 130–133 (1997).
[CrossRef]

Proc. Phys. Soc. (1)

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. 62, 405–417 (1949).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 13, pp. 629–633.

H. A. MacLeod, Thin Film Optical Filters, 2nd ed. (Hilger/McGraw-Hill, New York, 1986).
[CrossRef]

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4, p. 58.

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Figures (8)

Fig. 1
Fig. 1

FFP sensor, showing a single-mode fiber cavity coupled to an addressing fiber via a fusion-spliced titanium dioxide (TiO2) internal reflector. Lower-order multiple beam components are reflected and transmitted by input and end reflectors where, typically, n 0 = 1.465 (addressing fiber), n 1 = 1.465 (etalon fiber), and n 2 = 1.000 (air). The phase of the first reflected component a 1 is determined by reflection phase change β01 only. The phase of subsequent series a 2, a 3, … includes even-integer multiples of cavity phase delay ϕ.

Fig. 2
Fig. 2

Internal reflector phase changes showing (a) variation of τ1, β1, and χ = 2(β1 - τ1) with η = n 1 L/λ for a λ/4-thick TiO2 input reflector with losses of 25%, sandwiched between lossless dielectric fiber (n = 1.465) calculated from Ref. 13. (b) Variation of χ (max) with A 1 for a low-finesse device (R = 0.05) based on Eq. (4).

Fig. 3
Fig. 3

Limiting symmetric regimes of a FFP sensor in reflection. (a) Ideal lossless response A 1 = 0 (χ = π), R 1= R 2 = R = 0.1, 0.2, … 0.9. (b) Transmissionlike (high-loss) response A 1 = 0.56 (χ = 0), R 1 = 0.05, R 2 = 0.1, 0.2, … 0.9.

Fig. 4
Fig. 4

The asymmetric regime, for which 0 < χ < π, encompasses the broad range of possible field distributions between the strictly lossless and transmissionlike limits, and as such provides a more general description of the FFP sensor response. (a) Marked asymmetric response predicted for A 1 = 0.45, R1 = 0.12, R2 = 0.2, 0.3 … 0.9. (b) Transition from lossless (A 1 = 0, χ = π) through asymmetric (A 1 = 0.2, χ = 3π/4) to transmissionlike (A 1 = 0.67, χ = 0) response with increasing internal reflector losses.

Fig. 5
Fig. 5

System configuration for calibration and phase-stepped interrogation of FFP sensors, showing the current step (N = 3) applied for sensor demodulation. The current ramp is applied for calibration measurement of sensor transfer response. The sensor response is normalized with respect to the source intensity ramp (REF). SYNC, current step synchronization line; DAQ, data acquisition unit.

Fig. 6
Fig. 6

Measured transfer response of metal–dielectric FFP sensors obtained with the interferometer of Fig. 4 for (a) a MDD construction comprising an 80-nm aluminum end reflector, a 6-mm monomode fiber cavity, and a λ/4 TiO2 internal reflector (R 1/R 2 ≈ 0.15/0.75) and (b) a MDM construction comprising an 80-nm aluminum end reflector, a 20-mm monomode fiber cavity, and a 7-nm gold internal reflector (R 1/R 2 ≈ 0.15/0.75).

Fig. 7
Fig. 7

Systematic phase error of the cosine approximation [Eq. (6)] employed in three-step demodulation of a low-finesse (R = 0.05) FFP sensor with internal reflector losses [determined from Eq. (3) and (6)], showing cumulative error of the lossless + low-finesse approximation for internal reflector losses of a, 0%; b, 15%; and c, 25%.

Fig. 8
Fig. 8

Maximum phase error of the cosine (ΔϕCOS) and the lossless (ΔϕASY) approximations in the presence of higher-order cavity reflections and internal reflector losses determined from Eqs. (6) and (5) and Eqs. (3) and (5), respectively, for (a) a three-step algorithm (N = 3) and (b) a four-step algorithm (N = 4) with internal reflector losses of a, 5%; b, 15%; and c, 25%.

Equations (7)

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a1=r01 exp iβ01,  a2=t01t10r12 exp iτ01+τ10+β12+2ϕ,  a3=t01t10r10r122 exp iτ01+τ10+β10+2β12+2ϕ,
A=r01 exp iβ01+t01t10r12 exp i2τ+β12+2ϕ1-r10r12 exp iβ10+β12+2ϕ.
IRi=R1+R2T12+2T1R1R2cosψ-χ+δi-R1R2 cosχ+ϕi1+R1R2-2R1R2 cosψ+δi,
cosχmax/2=±A14R1T1
IRi=R1+R2-2R1R2 cosϕ+δi1+R1R2-2R1R2 cosϕ+δi=2R1-cosϕ+δi1+R2-2R cosϕ+δi,
IRi=KI01+V cosϕ+δi,
tan ϕ=2I2-I3-I1I3-I1,  N=3.

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