Abstract

In this paper, the design and the characterization of a novel interrogator based on integrated Fourier transform (FT) spectroscopy is presented. To the best of our knowledge, this is the first integrated FT spectrometer used for the interrogation of photonic sensors. It consists of a planar spatial heterodyne spectrometer, which is implemented using an array of Mach-Zehnder interferometers (MZIs) with different optical path differences. Each MZI employs a 3×3 multi-mode interferometer, allowing the retrieval of the complex Fourier coefficients. We derive a system of non-linear equations whose solution, which is obtained numerically from Newton’s method, gives the modulation of the sensor’s resonances as a function of time. By taking one of the sensors as a reference, to which no external excitation is applied and its temperature is kept constant, about 92% of the thermal induced phase drift of the integrated MZIs has been compensated. The minimum modulation amplitude that is obtained experimentally is 400 fm, which is more than two orders of magnitude smaller than the FT spectrometer resolution.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
  2. C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
    [Crossref]
  3. E. Hallynck and P. Bienstman, “Integrated optical pressure sensors in silicon-on-insulator,” IEEE Photonics J. 4, 443–450 (2012).
    [Crossref]
  4. K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
    [Crossref] [PubMed]
  5. X. Zhou, Y. Dai, J. M. Karanja, F. Liu, and M. Yang, “Microstructured FBG hydrogen sensor based on pt-loaded wo 3,” Opt. Express 25, 8777–8786 (2017).
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  6. Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
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  8. H. Li, X. Ma, B. Cui, Y. Wang, C. Zhang, J. Zhao, Z. Zhang, C. Tang, and E. Li, “Chip-scale demonstration of hybrid III – V / silicon photonic integration for an FBG interrogator,” Optica 4, 692–700 (2017).
    [Crossref]
  9. D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).
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    [Crossref]
  11. H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
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  12. U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
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  14. P. Orr and P. Niewczas, “High-speed, solid state, interferometric interrogator and multiplexer for fiber Bragg grating sensors,” J. Light. Technol. 29, 3387–3392 (2011).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
    [Crossref]
  22. A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
    [Crossref]
  23. D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  27. F. Soto-Eguibar and H. Moya-Cessa, “Inverse of the Vandermonde and Vandermonde confluent matrices,” Appl. Math. Inf. Sci. 5, 361–366 (2011).

2018 (3)

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

R. Uda, K. Yamaguchi, K. Takada, and K. Okamoto, “Fabrication of a silica-based complex fourier-transform integrated-optic spatial heterodyne spectrometer incorporating 120° optical hybrid couplers,” Appl. Opt. 57, 3781–3787 (2018).
[Crossref] [PubMed]

2017 (5)

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

H. Li, X. Ma, B. Cui, Y. Wang, C. Zhang, J. Zhao, Z. Zhang, C. Tang, and E. Li, “Chip-scale demonstration of hybrid III – V / silicon photonic integration for an FBG interrogator,” Optica 4, 692–700 (2017).
[Crossref]

F. G. Peternella, B. Ouyang, R. Horsten, M. Haverdings, P. Kat, and J. Caro, “Interrogation of a ring-resonator ultrasound sensor using a fiber mach-zehnder interferometer,” Opt. Express 25, 31622–31639 (2017).
[Crossref] [PubMed]

X. Zhou, Y. Dai, J. M. Karanja, F. Liu, and M. Yang, “Microstructured FBG hydrogen sensor based on pt-loaded wo 3,” Opt. Express 25, 8777–8786 (2017).
[Crossref] [PubMed]

2016 (1)

D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).

2015 (1)

C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
[Crossref]

2013 (4)

M. Perry, P. Orr, P. Niewczas, and M. Johnston, “High-speed interferometric fbg interrogator with dynamic and absolute wavelength measurement capability,” J. Light. Technol. 31, 2897–2903 (2013).
[Crossref]

H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
[Crossref]

U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

2012 (2)

V. M. N. Passaro, A. V. Tsarev, and F. De Leonardis, “Wavelength interrogator for optical sensors based on a novel thermo-optic tunable filter in SOI,” J. Light. Technol. 30, 2143–2150 (2012).
[Crossref]

E. Hallynck and P. Bienstman, “Integrated optical pressure sensors in silicon-on-insulator,” IEEE Photonics J. 4, 443–450 (2012).
[Crossref]

2011 (3)

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23, 1505–1507 (2011).
[Crossref]

P. Orr and P. Niewczas, “High-speed, solid state, interferometric interrogator and multiplexer for fiber Bragg grating sensors,” J. Light. Technol. 29, 3387–3392 (2011).
[Crossref]

F. Soto-Eguibar and H. Moya-Cessa, “Inverse of the Vandermonde and Vandermonde confluent matrices,” Appl. Math. Inf. Sci. 5, 361–366 (2011).

2010 (1)

K. Okamoto, H. Aoyagi, and K. Takada, “Fabrication of Fourier-transform, integrated-optic spatial heterodyne spectrometer on silica-based planar waveguide,” Opt. letters 35, 2103–2105 (2010).
[Crossref]

2009 (1)

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

2007 (1)

1999 (1)

K. B. Rochford and S. D. Dyer, “Demultiplexing of interferometrically interrogated fiber Bragg grating sensors using Hilbert transform processing,” J. Light. Technol. 17, 831–836 (1999).
[Crossref]

1996 (1)

A. D. Kersey, “A review of recent developments in fiber optic sensor technology,” Opt. Fiber Technol. 2, 291–317 (1996).
[Crossref]

1995 (1)

M. A. Davis and A. D. Kersey, “Application of a fiber fourier transform spectrometer to the detection of wavelength-encoded signals from bragg grating sensors,” J. Light. Technol. 13, 1289–1295 (1995).
[Crossref]

Aoyagi, H.

K. Okamoto, H. Aoyagi, and K. Takada, “Fabrication of Fourier-transform, integrated-optic spatial heterodyne spectrometer on silica-based planar waveguide,” Opt. letters 35, 2103–2105 (2010).
[Crossref]

Baets, R.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23, 1505–1507 (2011).
[Crossref]

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

Bienstman, P.

E. Hallynck and P. Bienstman, “Integrated optical pressure sensors in silicon-on-insulator,” IEEE Photonics J. 4, 443–450 (2012).
[Crossref]

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

Bock, P. J.

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Bogaerts, W.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23, 1505–1507 (2011).
[Crossref]

Bono, D.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Calvo, M. L.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Caro, J.

Cheben, P.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach-zehnder interferometers,” Opt. express 15, 18176–18189 (2007).
[Crossref]

Chen, S. L.

C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
[Crossref]

Coppola, G.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Corredera, P.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

Cui, B.

Dai, Y.

Dandridge, A.

A. Dandridge, “Fiber optic sensors based on the Mach-Zehnder and Michelson interferometers,” in Fiber Optic Sensors: an Introduction for engineers and scientists, E. Udd and W. B. Spillman, eds. (John Wiley and Sons, Inc., 1991), pp. 231–275.

Davis, M. A.

M. A. Davis and A. D. Kersey, “Application of a fiber fourier transform spectrometer to the detection of wavelength-encoded signals from bragg grating sensors,” J. Light. Technol. 13, 1289–1295 (1995).
[Crossref]

De Leonardis, F.

V. M. N. Passaro, A. V. Tsarev, and F. De Leonardis, “Wavelength interrogator for optical sensors based on a novel thermo-optic tunable filter in SOI,” J. Light. Technol. 30, 2143–2150 (2012).
[Crossref]

de Vos, K.

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

Delâge, A.

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Dyer, S. D.

K. B. Rochford and S. D. Dyer, “Demultiplexing of interferometrically interrogated fiber Bragg grating sensors using Hilbert transform processing,” J. Light. Technol. 17, 831–836 (1999).
[Crossref]

Favela, D.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Florjanczyk, M.

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach-zehnder interferometers,” Opt. express 15, 18176–18189 (2007).
[Crossref]

Ge, Y.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Girones, J.

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

Gu, T.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Guo, H.

H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
[Crossref]

Guo, L. J.

C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
[Crossref]

Hallynck, E.

E. Hallynck and P. Bienstman, “Integrated optical pressure sensors in silicon-on-insulator,” IEEE Photonics J. 4, 443–450 (2012).
[Crossref]

Haverdings, M.

Hens, Z.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23, 1505–1507 (2011).
[Crossref]

Herrero-Bermello, A.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

Horsten, R.

Hu, J.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Jain, S. C.

U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
[Crossref]

Jans, S.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

Janz, S.

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach-zehnder interferometers,” Opt. express 15, 18176–18189 (2007).
[Crossref]

Jin, J.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Johnston, M.

M. Perry, P. Orr, P. Niewczas, and M. Johnston, “High-speed interferometric fbg interrogator with dynamic and absolute wavelength measurement capability,” J. Light. Technol. 31, 2897–2903 (2013).
[Crossref]

Karanja, J. M.

Kat, P.

Kersey, A. D.

A. D. Kersey, “A review of recent developments in fiber optic sensor technology,” Opt. Fiber Technol. 2, 291–317 (1996).
[Crossref]

M. A. Davis and A. D. Kersey, “Application of a fiber fourier transform spectrometer to the detection of wavelength-encoded signals from bragg grating sensors,” J. Light. Technol. 13, 1289–1295 (1995).
[Crossref]

Kita, D. M.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Kleijn, E.

D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).

Lapointe, J.

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Lee, R.

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

Leijtens, X.

D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).

Li, E.

Li, H.

Liang, Q.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Lin, H.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Ling, T. L.

C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
[Crossref]

Liu, F.

Long, J.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Ma, X.

Michon, J.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Miranda, B.

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Moya-Cessa, H.

F. Soto-Eguibar and H. Moya-Cessa, “Inverse of the Vandermonde and Vandermonde confluent matrices,” Appl. Math. Inf. Sci. 5, 361–366 (2011).

Mrad, N.

H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
[Crossref]

Niewczas, P.

M. Perry, P. Orr, P. Niewczas, and M. Johnston, “High-speed interferometric fbg interrogator with dynamic and absolute wavelength measurement capability,” J. Light. Technol. 31, 2897–2903 (2013).
[Crossref]

P. Orr and P. Niewczas, “High-speed, solid state, interferometric interrogator and multiplexer for fiber Bragg grating sensors,” J. Light. Technol. 29, 3387–3392 (2011).
[Crossref]

Okamoto, K.

R. Uda, K. Yamaguchi, K. Takada, and K. Okamoto, “Fabrication of a silica-based complex fourier-transform integrated-optic spatial heterodyne spectrometer incorporating 120° optical hybrid couplers,” Appl. Opt. 57, 3781–3787 (2018).
[Crossref] [PubMed]

K. Okamoto, H. Aoyagi, and K. Takada, “Fabrication of Fourier-transform, integrated-optic spatial heterodyne spectrometer on silica-based planar waveguide,” Opt. letters 35, 2103–2105 (2010).
[Crossref]

K. Okamoto, “Fourier-transform, integrated-optic spatial heterodyne (fish) spectrometers on planar lightwave circuits,” in International Conference on Fibre Optics and Photonics, (Optical Society of America, 2012), p. M2A.1.
[Crossref]

Orr, P.

M. Perry, P. Orr, P. Niewczas, and M. Johnston, “High-speed interferometric fbg interrogator with dynamic and absolute wavelength measurement capability,” J. Light. Technol. 31, 2897–2903 (2013).
[Crossref]

P. Orr and P. Niewczas, “High-speed, solid state, interferometric interrogator and multiplexer for fiber Bragg grating sensors,” J. Light. Technol. 29, 3387–3392 (2011).
[Crossref]

Ouyang, B.

Passaro, V. M. N.

V. M. N. Passaro, A. V. Tsarev, and F. De Leonardis, “Wavelength interrogator for optical sensors based on a novel thermo-optic tunable filter in SOI,” J. Light. Technol. 30, 2143–2150 (2012).
[Crossref]

Perry, M.

M. Perry, P. Orr, P. Niewczas, and M. Johnston, “High-speed interferometric fbg interrogator with dynamic and absolute wavelength measurement capability,” J. Light. Technol. 31, 2897–2903 (2013).
[Crossref]

Peternella, F. G.

Podmore, H.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

Popelka, S.

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

Pustakhod, D.

D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).

Rochford, K. B.

K. B. Rochford and S. D. Dyer, “Demultiplexing of interferometrically interrogated fiber Bragg grating sensors using Hilbert transform processing,” J. Light. Technol. 17, 831–836 (1999).
[Crossref]

Schacht, E.

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

Schmid, J. H.

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Scott, A.

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach-zehnder interferometers,” Opt. express 15, 18176–18189 (2007).
[Crossref]

Shenoy, M. R.

U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
[Crossref]

Solheim, B.

Soto-Eguibar, F.

F. Soto-Eguibar and H. Moya-Cessa, “Inverse of the Vandermonde and Vandermonde confluent matrices,” Appl. Math. Inf. Sci. 5, 361–366 (2011).

Sun, W.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Takada, K.

R. Uda, K. Yamaguchi, K. Takada, and K. Okamoto, “Fabrication of a silica-based complex fourier-transform integrated-optic spatial heterodyne spectrometer incorporating 120° optical hybrid couplers,” Appl. Opt. 57, 3781–3787 (2018).
[Crossref] [PubMed]

K. Okamoto, H. Aoyagi, and K. Takada, “Fabrication of Fourier-transform, integrated-optic spatial heterodyne spectrometer on silica-based planar waveguide,” Opt. letters 35, 2103–2105 (2010).
[Crossref]

Tang, C.

Thyagarajan, K.

U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
[Crossref]

Tiwari, U.

U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
[Crossref]

Tsarev, A. V.

V. M. N. Passaro, A. V. Tsarev, and F. De Leonardis, “Wavelength interrogator for optical sensors based on a novel thermo-optic tunable filter in SOI,” J. Light. Technol. 30, 2143–2150 (2012).
[Crossref]

Uda, R.

Vachon, M.

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Velasco, A. V.

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

Wang, Y.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

H. Li, X. Ma, B. Cui, Y. Wang, C. Zhang, J. Zhao, Z. Zhang, C. Tang, and E. Li, “Chip-scale demonstration of hybrid III – V / silicon photonic integration for an FBG interrogator,” Optica 4, 692–700 (2017).
[Crossref]

Williams, K.

D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).

Xiao, G.

H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
[Crossref]

Xu, D. X.

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach-zehnder interferometers,” Opt. express 15, 18176–18189 (2007).
[Crossref]

Xu, D.-X.

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

Yamaguchi, K.

Yang, M.

Yao, J.

H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
[Crossref]

Yebo, N. A.

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23, 1505–1507 (2011).
[Crossref]

Zhang, C.

H. Li, X. Ma, B. Cui, Y. Wang, C. Zhang, J. Zhao, Z. Zhang, C. Tang, and E. Li, “Chip-scale demonstration of hybrid III – V / silicon photonic integration for an FBG interrogator,” Optica 4, 692–700 (2017).
[Crossref]

C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
[Crossref]

Zhang, D.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Zhang, Z.

Zhao, J.

Zhou, X.

Zou, K.

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

Appl. Math. Inf. Sci. (1)

F. Soto-Eguibar and H. Moya-Cessa, “Inverse of the Vandermonde and Vandermonde confluent matrices,” Appl. Math. Inf. Sci. 5, 361–366 (2011).

Appl. Opt. (1)

Biosens. Bioelectron. (1)

K. de Vos, J. Girones, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “SOI optical microring resonator with poly (ethylene glycol) polymer brush for label-free biosensor applications,” Biosens. Bioelectron. 24, 2528–2533 (2009).
[Crossref] [PubMed]

IEEE Photonics J. (1)

E. Hallynck and P. Bienstman, “Integrated optical pressure sensors in silicon-on-insulator,” IEEE Photonics J. 4, 443–450 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

N. A. Yebo, W. Bogaerts, Z. Hens, and R. Baets, “On-chip arrayed waveguide grating interrogated silicon-on-insulator microring resonator-based gas sensor,” IEEE Photonics Technol. Lett. 23, 1505–1507 (2011).
[Crossref]

IEEE Sensors J. (3)

D. Pustakhod, E. Kleijn, K. Williams, and X. Leijtens, “High-resolution awg-based fiber,” IEEE Sensors J. 28, 2203–2206 (2016).

Q. Liang, K. Zou, J. Long, J. Jin, D. Zhang, G. Coppola, W. Sun, Y. Wang, and Y. Ge, “Multi-component FBG-based force sensing systems by comparison with other sensing technologies : A review,” IEEE Sensors J. 18, 7345–7357 (2018).
[Crossref]

U. Tiwari, K. Thyagarajan, M. R. Shenoy, and S. C. Jain, “EDF-based edge-filter interrogation scheme for FBG sensors,” IEEE Sensors J. 13, 1315–1319 (2013).
[Crossref]

J. Light. Technol. (7)

V. M. N. Passaro, A. V. Tsarev, and F. De Leonardis, “Wavelength interrogator for optical sensors based on a novel thermo-optic tunable filter in SOI,” J. Light. Technol. 30, 2143–2150 (2012).
[Crossref]

P. Orr and P. Niewczas, “High-speed, solid state, interferometric interrogator and multiplexer for fiber Bragg grating sensors,” J. Light. Technol. 29, 3387–3392 (2011).
[Crossref]

M. Perry, P. Orr, P. Niewczas, and M. Johnston, “High-speed interferometric fbg interrogator with dynamic and absolute wavelength measurement capability,” J. Light. Technol. 31, 2897–2903 (2013).
[Crossref]

M. A. Davis and A. D. Kersey, “Application of a fiber fourier transform spectrometer to the detection of wavelength-encoded signals from bragg grating sensors,” J. Light. Technol. 13, 1289–1295 (1995).
[Crossref]

K. B. Rochford and S. D. Dyer, “Demultiplexing of interferometrically interrogated fiber Bragg grating sensors using Hilbert transform processing,” J. Light. Technol. 17, 831–836 (1999).
[Crossref]

C. Zhang, S. L. Chen, T. L. Ling, and L. J. Guo, “Imprinted polymer microrings as high performance ultrasound detectors in photoacoustic imaging,” J. Light. Technol. 33, 4318–4328 (2015).
[Crossref]

H. Guo, G. Xiao, N. Mrad, and J. Yao, “Echelle diffractive grating based wavelength interrogator for potential aerospace applications,” J. Light. Technol. 31, 2099–2105 (2013).
[Crossref]

Nat. Commun. (1)

D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Fiber Technol. (1)

A. D. Kersey, “A review of recent developments in fiber optic sensor technology,” Opt. Fiber Technol. 2, 291–317 (1996).
[Crossref]

Opt. letters (4)

K. Okamoto, H. Aoyagi, and K. Takada, “Fabrication of Fourier-transform, integrated-optic spatial heterodyne spectrometer on silica-based planar waveguide,” Opt. letters 35, 2103–2105 (2010).
[Crossref]

A. V. Velasco, P. Cheben, P. J. Bock, A. Delâge, J. H. Schmid, J. Lapointe, S. Janz, M. L. Calvo, D. X. Xu, M. Florjańczyk, and M. Vachon, “High-resolution fourier-transform spectrometer chip with microphotonic silicon spiral waveguides,” Opt. letters 38, 706–708 (2013).
[Crossref]

H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing fourier-transform on-chip spectrometer,” Opt. letters 42, 1440–1443 (2017).
[Crossref]

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Jans, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. letters 42, 2239–2242 (2017).
[Crossref]

Optica (1)

Other (2)

K. Okamoto, “Fourier-transform, integrated-optic spatial heterodyne (fish) spectrometers on planar lightwave circuits,” in International Conference on Fibre Optics and Photonics, (Optical Society of America, 2012), p. M2A.1.
[Crossref]

A. Dandridge, “Fiber optic sensors based on the Mach-Zehnder and Michelson interferometers,” in Fiber Optic Sensors: an Introduction for engineers and scientists, E. Udd and W. B. Spillman, eds. (John Wiley and Sons, Inc., 1991), pp. 231–275.

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Figures (6)

Fig. 1
Fig. 1 (a) Picture of the FT spectrometer chip. ΔLm is given by ΔLm = mΔL1 with ΔL1 = 0.710 mm, leading to Fm = F1/m, where m is an integer number ranging from 1 to 9. The different MZIs are identified with the index m. The 2×2 MMIs are indicated in white. All other power splitters are 1×2 MMIs. (b) Cross-section of the shallow etch waveguide. The refractive indexes at the wavelength of 1550 nm are also indicated. (c) Cross-section of the deep etch waveguide. (d) Schematic of an optical fiber aligned to one of the inputs of the chip. For input #4, θwg = θf = 0°. For all other inputs, θwg = 7° and θf = 23°. (e) Schematic of the FT spectrometer and the PCB that implements the TIAs and a pre-processing module. The outputs are sampled by the DAQ.
Fig. 2
Fig. 2 (a) Traces of V1,x and V1,y as a function of the laser wavelength. We fitted Eq. (6) against the data points and we obtained F1 = 921.7 ± 0.5 pm, δϕ1 = 17.9 ±0.3°, A1,x = 1.449 ± 0.003V and A1,y = 1.234 ± 0.004V. (b) Lissajous plot of the data points [V1,x(λ), V1,y(λ)] shown in Fig. 2(a). By fitting an ellipse to the data points we got 1.56 V and 1.09 V for the semi-axis values and 31.2° for the tilt angle with respect to the x-axis.
Fig. 3
Fig. 3 Illustration of the calibration procedure for two sensors. (a) Independent excitation of sensor 1 and sensor 2. (b) Simulated values of V1,x(t) and V1,y(t) for MZI 1. The changes in time of the functions V1,x(t) and V1,y(t) are caused by the modulation of the peak wavelengths shown in Fig. 3(a). The voltages Vm,x(t) and Vm,y(t) (m = 1,...,M) are measured by our acquisition system. Vm,x(t) and Vm,y(t) are obtained from Eq. (27). For this simulation, Vm,x(t) = Vm,x(t) and Vm,y(t) = Vm,y(t). (c) Lissajous curve (V1,x(t), V1,y(t)) for MZI 1. The modulation of the peak wavelength of the sensors induces an angular deflection in the plane of the voltages V1,x and V1,y. From the Lissajous curve, the complex modulus and the phase of the coefficients amk were extracted. For this simulation, F1 =1.0 nm.
Fig. 4
Fig. 4 (a) Schematic of the setup. Light from an ASE source is sent, through a circulator, to the FBG sensor array. The FBG sensors reflect back to the circulator their combined spectrum, which is amplified by an optical booster amplifier (gain = 12dB). Light is coupled to the chip using lensed fibers. (b) Schematic of the temperature / strain sensors. 0 = 1.74 m, which is the fiber length between the clamps.
Fig. 5
Fig. 5 Main results of the interrogation. (a) Time traces of the real and imaginary parts of 1(t). A low pass filter (cut-off at 45 Hz) has been applied to the measured voltages Vm,x(t) and Vm,y(t). The numbers 1,2,3 and 4 indicate the calibration interval ( t k start < t < t k end) for sensors k =1,...,4. (b) Lissajous plot obtained by plotting the real and imaginary parts of 1(t). During the calibration, the Lissajous curve is a circular arc. During the interrogation, all sensors are simultaneously excited, and an arbitrary Lissajous curve is obtained as shown in orange. (c)–(e) Solutions δ2(t), δ3(t) and δ4(t) of Eq. (36) for t > 0. FBG #4 is the reference sensor. The phase drift was compensated using Eq. (39). (f) Comparison between the solutions δ1(t) and δ1(t). The inset shows a zoom of the solution δ1(t).
Fig. 6
Fig. 6 Modulation amplitude Δλ(1) of sensor 1 as a function of the strain applied. Δλ(1) is calculated from as Δ λ ( 1 ) = | δ 1 , 3 j dip ¯ δ 1 , 3 j max ¯ |, where δ 1 , 3 j dip ¯ and δ 1 , 3 j max ¯ are defined in upper inset of the fig. A straight line has been fitted to the data points ( | ε j ( 1 ) |, Δ λ j ( 1 )). The slope, whose value is 1.217±0.006 pm/microstrain, gives the sensitivity of FBG #1. The inset in the bottom of the fig. shows the data points ( ε j ( 1 ), Δ λ j ( 1 )) and the straight line fitted in a Loglog plot. The minimum amplitude modulation retrieved is 400±200 pm.

Equations (42)

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T ml ( λ ) = 1 3 [ 1 + v ml cos ( 2 π n eff , m ( λ ) Δ L m λ + ϕ l ) ] ,
n eff , m ( λ ) λ n eff ( λ 0 ) + n g + δ n eff , m λ 0 n g λ 0 2 λ ,
T ml ( λ ) = 1 3 [ 1 + v ml cos [ 2 π λ F m ϕ l Ψ m ) ] ,
Ψ m = 2 π Δ L m λ 0 ( n g + n eff ( λ 0 ) + δ n eff , m ) .
V ml ( λ ) = g ml P m R pm T m k ( λ ) = g ml P m R p m 3 [ 1 + v ml cos ( 2 π m λ F 1 ϕ l Ψ m ) ] ,
V m , x ( λ ) = 2 V m , 3 V m , 1 V m , 2 = A m , x cos ( 2 π m λ F 1 Ψ m ) + x off , m , V m , y ( λ ) = 3 ( V m , 2 V m , 3 ) = A m , y sin ( 2 π m λ F 1 Ψ m δ ϕ m ) + y off , m ,
S ( λ , λ 1 ( t ) , , λ K ( t ) ) = k = 1 K s k ( λ , λ k ( t ) ) = k = 1 K s k ( λ λ k ( t ) ) ,
V ml ( t ) = G S ( λ , λ 1 ( t ) , , λ K ( t ) ) T ml ( λ ) d λ ,
V m , x ( t ) = 3 G S ( λ , λ 1 ( t ) , , λ K ( t ) ) cos ( 2 π m λ F 1 Ψ m ) d λ + x off , m ,
V m , y ( t ) = 3 G S ( λ , λ 1 ( t ) , , λ K ( t ) ) sin ( 2 π m λ F 1 Ψ m ) d λ + y off , m .
V ^ m ( t ) = 3 G e i Ψ m S ( λ , λ 1 ( t ) , , λ K ( t ) ) exp ( i 2 π m F 1 λ ) d λ .
V ^ m ( t ) e i Ψ m 3 G = λ 0 F 1 / 2 λ 0 + F 1 / 2 S ( λ , λ 1 ( t ) , , λ K ( t ) ) exp ( i 2 π m F 1 λ ) d λ .
S ( λ , λ 1 ( t ) , , λ K ( t ) ) = 1 3 G m = V ^ m ( t ) e i Ψ m exp ( i 2 π m F 1 λ ) = 2 3 G m = 0 [ V m , x ( t ) cos ( 2 π m F 1 λ Ψ m ) V m , y ( t ) sin ( 2 π m F 1 λ Ψ m ) ] ,
S M ( λ , λ 1 ( t ) , , λ K ( t ) ) = 2 3 G m = 0 M [ V m , x ( t ) cos ( 2 π m F 1 λ Ψ m ) V m , y ( t ) sin ( 2 π m F 1 λ Ψ m ) ] .
δ λ res = F 1 2 M .
λ k ( t ) = λ k ( 0 ) + δ k ( t ) ,
V ^ m ( t ) = 3 G e i Ψ m k = 1 K s k ( λ λ k ( 0 ) δ k ( t ) ) exp ( i 2 π m F 1 λ ) d λ .
V ^ m ( t ) = 3 G k = 1 K s ^ k ( m / F 1 ) exp [ i ( Ψ m + 2 π m F 1 λ k ( 0 ) ) ] exp ( i 2 π m F 1 δ k ( t ) ) ,
a m k = 3 G s ^ k ( m / F 1 ) exp [ i ( Ψ m + 2 π m F 1 λ k ( 0 ) ) ] .
V ^ m ( t ) = k = 1 K a m k exp [ ( i 2 π F 1 δ k ( t ) ) m ] ,
V ^ 1 ( t ) = a 11 exp [ i 2 π δ 1 ( t ) F 1 ] + a 12 exp [ i 2 π δ 2 ( t ) F 1 ] + + a 1 K exp [ i 2 π δ K ( t ) F 1 ] , V ^ 2 ( t ) = a 21 exp [ 2 i 2 π δ 1 ( t ) F 1 ] + a 22 exp [ 2 i 2 π δ 2 ( t ) F 1 ] + + a 2 K exp [ 2 i 2 π δ K ( t ) F 1 ] , V ^ M ( t ) = a M 1 exp [ M i 2 π δ 1 ( t ) F 1 ] + a M 2 exp [ M i 2 π δ 2 ( t ) F 1 ] + + a M K exp [ M i 2 π δ K ( t ) F 1 ] .
a m k = 3 G s k max 2 exp ( m O P D 1 L c , k ) exp [ i ( Ψ m + 2 π m F 1 λ k ( 0 ) ) ] ,
L c , k = λ 0 2 π w k ,
V ^ m ( t ) = a m k e i 2 π m F 1 δ k ( t ) + l k K a ml = | a m k | e i θ m k ( t ) + c m k ,
θ m k ( t ) = m 2 π δ k ( t ) F 1 + arg ( a m k ) .
( V m , x ( t ) , V m , y ( t ) ) | t k start < t < t k end = ( { V ^ m ( t ) } , { V ^ m ( t ) } ) | t k start < t < t k end = [ | a m k | cos ( θ m k ( t ) ) + { c m k } , | a m k | sin ( ( θ m k ( t ) ) + { c m k } ] | t k start < t < t k end ,
( V m , x ( t ) V m , y ( t ) ) = ( r 1 , m k / r 2 , m k 0 0 1 ) ( cos α sin α sin α cos α ) ( V m , x ( t ) V m , y ( t ) ) ,
| a m k | = r 1 , m k .
θ m k ( t ) = arctan 2 ( V y , m ( t ) { c m , k } , V x , m ( t ) { c m , k } ) ,
θ m k ( t k end ) = m 2 π δ k ( t k end ) F 1 + arg ( a m k ) = m 2 π δ k ( 0 ) F 1 + arg ( a m k ) ,
arg ( a m k ) = θ m k ( t k end ) = θ m k ( 0 ) .
x off , m = 1 | t 0 | t 0 t { V m , y ( t ) k | a m k | cos [ θ m k ( t ) ] } d t , y off , m = 1 | t 0 | t 0 t { V m , y ( t ) k | a m k | sin [ θ m k ( t ) ] } d t .
V ^ m ( t ) = [ V x , m ( t ) x off , m ] + i [ V y , m ( t ) y off , m ] .
Ψ m ( t ) = m 2 π Δ L λ 0 ( n g + n eff ( λ 0 ) ( T 0 ) + n eff T Δ T ( t ) + δ n eff , m ) = Ψ m ( 0 ) + m Δ Ψ ( t ) ,
Δ Ψ ( t ) = 2 π Δ L λ 0 n eff T Δ T ( t ) .
V ^ m ( t ) = 3 G k = 1 K s ^ k ( m / F 1 ) exp [ i ( Ψ m ( 0 ) + 2 π λ k ( 0 ) F 1 ) ] exp [ i 2 π m F 1 ( δ k ( t ) Δ Ψ ( t ) F 1 2 π ) ] = m = 1 M a m k exp [ ( i 2 π m F 1 δ k ( t ) ) m ] ,
δ k ( t ) = δ k ( t ) Δ Ψ ( t ) F 1 / ( 2 π ) .
δ ref ( t ) = Δ Ψ ( t ) F 1 / ( 2 π ) .
δ k ( t ) = δ k ( t ) δ ref ( t ) .
ε j ( 1 ) = Δ 3 j ( 1 ) 0 ,
Δ λ j ( 1 ) = | δ 1 , 3 j dip ¯ δ 1 , 3 j max ¯ | ,
S ( 1 ) = Δ λ ( 1 ) ε ( 1 ) .