Abstract

A CWDM network operating in reflective configuration for multiplexing remote Radio-Frequency (RF) self-referenced fiber-optic intensity sensors is analyzed and experimentally investigated. In the described approach, the use of fiber Bragg gratings as spectral selective mirrors allows to implement delay lines in the electrical domain, achieving more compact sensor-heads and easy-reconfigurable sensing points. Two measurement parameters for the sensing heads are defined and comparatively studied in terms of design parameters, linearity, sensitivity and resolution. The proposed sensor configuration is modeled following the Z-transform formalism, which permits an easy analysis of the system frequency response. Experimental results are presented, showing the characterization of the network performance and considering the properties of sensor self-referencing as well as sensor crosstalk.

© 2010 OSA

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  1. O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
    [CrossRef]
  2. Y. Zhu and A. Wang, “Miniature fibre-optic pressure sensor,” IEEE Photon. Technol. Lett. 17(2), 447–449 (2005).
    [CrossRef]
  3. J. M. Corres, F. J. Arregui, and I. R. Matias, “Design of humidity sensors based on tapered optical fibres,” J. Lightwave Technol. 24(11), 4329–4336 (2006).
    [CrossRef]
  4. J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
    [CrossRef]
  5. J. Montalvo, C. Vázquez, and D. S. Montero, “CWDM self-referencing sensor network based on ring resonators in reflective configuration,” Opt. Express 14(11), 4601–4610 (2006).
    [CrossRef] [PubMed]
  6. J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
    [CrossRef]
  7. D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
    [CrossRef]
  8. J. M. Baptista, J. L. Santos, and A. S. Lage, “Mach-Zender and Michelson topologies for self-referencing fiber-optic intensity sensors,” Opt. Eng. 39(6), 1636–1644 (2000).
    [CrossRef]
  9. J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
    [CrossRef]
  10. J. M. Baptista, J. L. Santos, and A. S. Lage, “Self-referenced fibre optic intensity sensor based on a multiple beam Sagnac topology,” Opt. Commun. 181(4-6), 287–294 (2000).
    [CrossRef]
  11. C. Vázquez, J. Montalvo, and P. C. Lallana, “Radio-frequency ring resonators for self-referencing fiber-optic intensity sensors,” Opt. Eng. Lett. 44, 1–2 (2005).
  12. C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
    [CrossRef]
  13. S. Abad, M. López-Amo, F. M. Araújo, L. A. Ferreira, and J. L. Santos, “Fiber Bragg grating-based self-referencing technique for wavelength-multiplexed intensity sensors,” Opt. Lett. 27(4), 222–224 (2002).
    [CrossRef]
  14. J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
    [CrossRef]

2009 (2)

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
[CrossRef]

2008 (1)

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

2006 (5)

J. Montalvo, C. Vázquez, and D. S. Montero, “CWDM self-referencing sensor network based on ring resonators in reflective configuration,” Opt. Express 14(11), 4601–4610 (2006).
[CrossRef] [PubMed]

J. M. Corres, F. J. Arregui, and I. R. Matias, “Design of humidity sensors based on tapered optical fibres,” J. Lightwave Technol. 24(11), 4329–4336 (2006).
[CrossRef]

C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
[CrossRef]

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

2005 (2)

C. Vázquez, J. Montalvo, and P. C. Lallana, “Radio-frequency ring resonators for self-referencing fiber-optic intensity sensors,” Opt. Eng. Lett. 44, 1–2 (2005).

Y. Zhu and A. Wang, “Miniature fibre-optic pressure sensor,” IEEE Photon. Technol. Lett. 17(2), 447–449 (2005).
[CrossRef]

2004 (1)

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

2002 (1)

2000 (2)

J. M. Baptista, J. L. Santos, and A. S. Lage, “Self-referenced fibre optic intensity sensor based on a multiple beam Sagnac topology,” Opt. Commun. 181(4-6), 287–294 (2000).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Mach-Zender and Michelson topologies for self-referencing fiber-optic intensity sensors,” Opt. Eng. 39(6), 1636–1644 (2000).
[CrossRef]

Abad, S.

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

S. Abad, M. López-Amo, F. M. Araújo, L. A. Ferreira, and J. L. Santos, “Fiber Bragg grating-based self-referencing technique for wavelength-multiplexed intensity sensors,” Opt. Lett. 27(4), 222–224 (2002).
[CrossRef]

Araújo, F. M.

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

S. Abad, M. López-Amo, F. M. Araújo, L. A. Ferreira, and J. L. Santos, “Fiber Bragg grating-based self-referencing technique for wavelength-multiplexed intensity sensors,” Opt. Lett. 27(4), 222–224 (2002).
[CrossRef]

Arregui, F. J.

Arrúe, J.

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

Baptista, J. M.

J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
[CrossRef]

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Self-referenced fibre optic intensity sensor based on a multiple beam Sagnac topology,” Opt. Commun. 181(4-6), 287–294 (2000).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Mach-Zender and Michelson topologies for self-referencing fiber-optic intensity sensors,” Opt. Eng. 39(6), 1636–1644 (2000).
[CrossRef]

Corres, J. M.

Ferreira, L. A.

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

S. Abad, M. López-Amo, F. M. Araújo, L. A. Ferreira, and J. L. Santos, “Fiber Bragg grating-based self-referencing technique for wavelength-multiplexed intensity sensors,” Opt. Lett. 27(4), 222–224 (2002).
[CrossRef]

Frazao, O.

J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
[CrossRef]

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

Jäger, D.

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

Lage, A. S.

J. M. Baptista, J. L. Santos, and A. S. Lage, “Self-referenced fibre optic intensity sensor based on a multiple beam Sagnac topology,” Opt. Commun. 181(4-6), 287–294 (2000).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Mach-Zender and Michelson topologies for self-referencing fiber-optic intensity sensors,” Opt. Eng. 39(6), 1636–1644 (2000).
[CrossRef]

Lallana, P. C.

C. Vázquez, J. Montalvo, and P. C. Lallana, “Radio-frequency ring resonators for self-referencing fiber-optic intensity sensors,” Opt. Eng. Lett. 44, 1–2 (2005).

López-Amo, M.

Marques, L. M.

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

Matias, I. R.

Möllers, I.

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

Montalvo, J.

J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
[CrossRef]

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

J. Montalvo, C. Vázquez, and D. S. Montero, “CWDM self-referencing sensor network based on ring resonators in reflective configuration,” Opt. Express 14(11), 4601–4610 (2006).
[CrossRef] [PubMed]

C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
[CrossRef]

C. Vázquez, J. Montalvo, and P. C. Lallana, “Radio-frequency ring resonators for self-referencing fiber-optic intensity sensors,” Opt. Eng. Lett. 44, 1–2 (2005).

Montero, D. S.

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

J. Montalvo, C. Vázquez, and D. S. Montero, “CWDM self-referencing sensor network based on ring resonators in reflective configuration,” Opt. Express 14(11), 4601–4610 (2006).
[CrossRef] [PubMed]

C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
[CrossRef]

Pena, J. M. S.

C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
[CrossRef]

Rego, G.

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

Rego, G. M.

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

Santos, J. L.

J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
[CrossRef]

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

S. Abad, M. López-Amo, F. M. Araújo, L. A. Ferreira, and J. L. Santos, “Fiber Bragg grating-based self-referencing technique for wavelength-multiplexed intensity sensors,” Opt. Lett. 27(4), 222–224 (2002).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Self-referenced fibre optic intensity sensor based on a multiple beam Sagnac topology,” Opt. Commun. 181(4-6), 287–294 (2000).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Mach-Zender and Michelson topologies for self-referencing fiber-optic intensity sensors,” Opt. Eng. 39(6), 1636–1644 (2000).
[CrossRef]

Santos, S.

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

Santos, S. F.

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

Vázquez, C.

J. Montalvo, O. Frazao, J. L. Santos, C. Vázquez, and J. M. Baptista, “Radio-frequency self-referencing technique with enhanced sensitivity for coarse WDM fiber optic intensity sensors,” J. Lightwave Technol. 27(5), 475–482 (2009).
[CrossRef]

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

J. Montalvo, C. Vázquez, and D. S. Montero, “CWDM self-referencing sensor network based on ring resonators in reflective configuration,” Opt. Express 14(11), 4601–4610 (2006).
[CrossRef] [PubMed]

C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
[CrossRef]

C. Vázquez, J. Montalvo, and P. C. Lallana, “Radio-frequency ring resonators for self-referencing fiber-optic intensity sensors,” Opt. Eng. Lett. 44, 1–2 (2005).

Wang, A.

Y. Zhu and A. Wang, “Miniature fibre-optic pressure sensor,” IEEE Photon. Technol. Lett. 17(2), 447–449 (2005).
[CrossRef]

Zhu, Y.

Y. Zhu and A. Wang, “Miniature fibre-optic pressure sensor,” IEEE Photon. Technol. Lett. 17(2), 447–449 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

O. Frazao, L. M. Marques, S. Santos, J. M. Baptista, and J. L. Santos, “Simultaneous measurement for strain and temperature based on a long-period grating combined with a high-birefrigence fiber loop mirror,” IEEE Photon. Technol. Lett. 18(22), 2407–2409 (2006).
[CrossRef]

Y. Zhu and A. Wang, “Miniature fibre-optic pressure sensor,” IEEE Photon. Technol. Lett. 17(2), 447–449 (2005).
[CrossRef]

C. Vázquez, J. Montalvo, D. S. Montero, and J. M. S. Pena, “Self-referencing fiber-optic intensity sensors using ring resonators and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(22), 2374–2376 (2006).
[CrossRef]

J. Montalvo, F. M. Araújo, L. A. Ferreira, C. Vázquez, and J. M. Baptista, “Electrical FIR filter with optical coefficients for self-referencing WDM intensity sensors,” IEEE Photon. Technol. Lett. 20(1), 45–47 (2008).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Commun. (2)

J. M. Baptista, S. F. Santos, G. Rego, O. Frazao, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006).
[CrossRef]

J. M. Baptista, J. L. Santos, and A. S. Lage, “Self-referenced fibre optic intensity sensor based on a multiple beam Sagnac topology,” Opt. Commun. 181(4-6), 287–294 (2000).
[CrossRef]

Opt. Eng. (2)

J. M. Baptista, J. L. Santos, and A. S. Lage, “Mach-Zender and Michelson topologies for self-referencing fiber-optic intensity sensors,” Opt. Eng. 39(6), 1636–1644 (2000).
[CrossRef]

J. M. Baptista, S. Abad, G. M. Rego, L. A. Ferreira, and J. L. Santos, “Wavelength multiplexing of frequency- based self-referenced fiber optic intensity sensors,” Opt. Eng. 43(3), 702–707 (2004).
[CrossRef]

Opt. Eng. Lett. (1)

C. Vázquez, J. Montalvo, and P. C. Lallana, “Radio-frequency ring resonators for self-referencing fiber-optic intensity sensors,” Opt. Eng. Lett. 44, 1–2 (2005).

Opt. Express (1)

Opt. Lett. (1)

Sensors (1)

D. S. Montero, C. Vázquez, I. Möllers, J. Arrúe, and D. Jäger, “A self-referencing intensity based polymer optical fiber sensor for liquid detection,” Sensors 9(8), 6446–6455 (2009).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic of the proposed electro-optical CWDM network for supporting N self-referenced optical fiber intensity sensors (FOSi, i = 1,…,N). BLS: Broadband Light Source, IM: Intensity Modulator, PD: Photodetector, FOS: Fiber Optic Sensor.

Fig. 2
Fig. 2

(a) Point-to-point self-electro-optical configuration for a generic remote sensing point. (b) Filter model of the configuration for a remote sensing point with two electrical phase-shifts at the reception stage.

Fig. 3
Fig. 3

Normalized magnitude response (a) and phase response (b) of the transfer function of the self-referencing configuration versus angular frequency for different values of β.

Fig. 4
Fig. 4

Theoretical curves of the R i parameter (a) and the output phase ϕ i (b) versus β i for different phase-shifts configurations at the reception stage.

Fig. 5
Fig. 5

Calibration curves and hysteresis of the sensor loss modulations H1 (a) and H2 (b) for the taper based displacement sensors.

Fig. 6
Fig. 6

(a) Self-reference test of R 1 versus power fluctuations up to 12dB for different values of β 1 and β 2 = 0.25 . (b) Self-reference test of ϕ 1 versus power fluctuations up to 12dB for different values of β 1 and β 2 = 0.25 .

Fig. 7
Fig. 7

Measurements of the R 1 parameter (a) and the output phase ϕ 1 (b) versus β 1 (FOS1) for different values of β 2 (FOS2).

Fig. 8
Fig. 8

(a) S R versus β for different values of the phase-shifts at the reception stage with Ω > 0 . (b) S R versus β for different phase-shifts at the reception stage with Ω < 0 . In both cases dashed lines were obtained from the theory.

Fig. 9
Fig. 9

(a) Theoretical sensitivity (absolute value) of the output phase ϕ versus β for different values of Ω (b) Measurements of the sensitivity S ϕ of the output phase versus β for different phase-shift configurations at the reception stage (dashed lines give the values obtained from the theory).

Fig. 10
Fig. 10

Measurement of R versus β when ( Ω 1 , Ω 2 ) = ( 104 º , 180 º ) is applied at the reception stage (linear regression in solid line and theoretical curve in dashed line).

Fig. 11
Fig. 11

Measurement of ϕ versus β when ( Ω 1 , Ω 2 ) = ( 0 º , 120 º ) is applied at the reception stage (linear regression in solid line and theoretical curve in dashed line).

Tables (1)

Tables Icon

Table 1 Comparative of R and ϕ parameters in terms of resolution.

Equations (20)

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

p 0 ( t ) = α i ( p R i ( t ) + β i ( t ) p S i ( t ) )
α i = m R i . R ( λ R i ) . d R i
β i = m S i . R ( λ S i ) . d S i m R i . R ( λ R i ) . d R i H i 2
P R i = P i n α i exp ( j Ω 1 ) P S i = P i n α i β i exp ( j Ω 2 )
H 0 = P 0 P i n = α i ' [ 1 + β i exp [ j ( Ω 2 Ω 1 ) ] ]
H 0 ( z ) = α i ' ( 1 + β i z 1 )
R i = V O ( f , Ω 2 ) V O ( f , Ω 1 ) = M ( f , Ω 2 ) | Ω 1 = 0 M ( f , Ω 1 ) | Ω 2 = 0 = [ 1 + ( 2 β 1 + β 2 ) cos Ω 2 ] 1 / 2 [ 1 + ( 2 β 1 + β 2 ) cos Ω 1 ] 1 / 2
M ( f , Ω 1 , Ω 2 ) = α i ( 1 + 2 β i cos Ω i + β i 2 ) 1 / 2
ϕ i = arctan [ ( sin Ω 1 + β i sin Ω 2 ) ( cos Ω 1 + β i cos Ω 2 ) ]
R i > 1
m ( ϕ i ) > 0
Ω 1 Ω 2 > 0
R i < 1
m ( ϕ i ) < 0
Ω 1 Ω 2 < 0
S R ( Ω 1 , Ω 2 , β ) = ( cos Ω 2 cos Ω 1 ) [ 1 β 2 ] ( 1 + 2 β cos Ω 1 + β 2 ) 3 / 2 ( 1 + 2 β cos Ω 2 + β 2 ) 1 / 2
S ϕ ( Ω 1 , Ω 2 , β ) = sin ( Ω 1 Ω 2 ) 1 + 2 β cos ( Ω 1 Ω 2 ) + β 2
Δ β min = Δ ϕ min S ϕ
Δ β min = Δ R min S R
Δ R min = ( Δ M 1 M 1 + Δ M 2 M 2 ) M 1 M 2 = Δ M ( 1 M 1 + M 1 M 2 2 ) Δ M M 2

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