Abstract

In this paper, we report on theoretical investigation of split mode resonant sensors based on fiber Bragg grating (FBG) ring resonators and π-shifted fiber Bragg grating (π-FBG) ring resonators. By using a π-shifted Bragg grating ring resonator (π-FBGRR) instead of a conventional fiber Bragg grating ring resonator (FBGRR), the symmetric and antisymmetric resonance branches (i.e., the eigen-modes of the perturbed system) show peculiar and very important features that can be exploited to improve the performance of the fiber optic spectroscopic sensors. In particular, the π-FBGRR symmetric resonance branch can be taylored to have a maximum splitting sensitivity to small environmental perturbations. This optimal condition has been found around the crossing points of the two asymmetric resonance branches, by properly choosing the physical parameters of the system. Then, high sensitivity splitting mode sensors are theoretically demonstrated showing, as an example, a strain sensitivity improvement of at least one order of magnitude over the state-of-the-art.

© 2014 Optical Society of America

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References

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2014 (3)

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

C. E. Campanella, C. M. Campanella, F. De Leonardis, and V. M. N. Passaro, “A high efficiency label-free photonic biosensor based on vertically stacked ring resonators,” Eur. Phys. J. Spec. Top. 223, 1–13 (2014).

X. Bai, D. Fan, S. Wang, S. Pu, and X. Zeng, “Strain Sensor Based on Fiber Ring Cavity Laser With Photonic Crystal Fiber In-Line Mach–Zehnder Interferometer,” IEEE Photon. J. 6(4), 6801608 (2014).
[Crossref]

2013 (4)

K. K. Qureshi, Z. Y. Liu, H. Y. Tam, and M. F. Zia, “A strain sensor based on in-line fiber Mach–Zehnder interferometer in twin-core photonic crystal fiber,” Opt. Commun. 309(15), 68–70 (2013).
[Crossref]

J. R. Zheng, P. Yan, Y. Yu, Z. Ou, J. Wang, X. Chen, and C. Du, “Temperature and index insensitive strain sensor based on a photonic crystal fiber in line Mach– Zehnder interferometer,” Opt. Commun. 297(15), 7–11 (2013).
[Crossref]

M. Li, X. Wu, L. Liu, X. Fan, and L. Xu, “Self-Referencing Optofluidic Ring Resonator Sensor for Highly Sensitive Biomolecular Detection,” Anal. Chem. 85(19), 9328–9332 (2013).
[Crossref] [PubMed]

C. E. Campanella, A. Giorgini, S. Avino, P. Malara, R. Zullo, G. Gagliardi, and P. De Natale, “Localized strain sensing with fiber Bragg-grating ring cavities,” Opt. Express 21(24), 29435–29441 (2013).
[Crossref] [PubMed]

2012 (1)

S. N. Chormaic, Y. Wu, and J. M. Ward, “Whispering gallery mode resonators as tools for non-linear optics and optomechanics,” Proc. SPIE 8236, 82361K (2012).
[Crossref]

2011 (1)

2010 (3)

2007 (2)

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of Macrobending Losses of Standard Single Mode Fiber with Small Bend Radii,” Microw. Opt. Technol. Lett. 49(9), 2133–2138 (2007).
[Crossref]

H. Y. Choi, M. J. Kim, and B. H. Lee, “All-fiber Mach-Zehnder type interferometers formed in photonic crystal fiber,” Opt. Express 15(9), 5711–5720 (2007).
[Crossref] [PubMed]

2006 (2)

J. Ctyroky, I. Richter, and M. Sinor, “Dual resonance in a waveguide coupled ring micro-resonator,” Opt. Quantum Electron. 38(9–11), 781–797 (2006).

W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
[Crossref] [PubMed]

2005 (2)

A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3), 036619 (2005).
[Crossref] [PubMed]

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

2002 (1)

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

1997 (1)

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Armenise, M. N.

Avino, S.

Bai, X.

X. Bai, D. Fan, S. Wang, S. Pu, and X. Zeng, “Strain Sensor Based on Fiber Ring Cavity Laser With Photonic Crystal Fiber In-Line Mach–Zehnder Interferometer,” IEEE Photon. J. 6(4), 6801608 (2014).
[Crossref]

Campanella, C. E.

Campanella, C. M.

C. E. Campanella, C. M. Campanella, F. De Leonardis, and V. M. N. Passaro, “A high efficiency label-free photonic biosensor based on vertically stacked ring resonators,” Eur. Phys. J. Spec. Top. 223, 1–13 (2014).

Cassidy, J.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of Macrobending Losses of Standard Single Mode Fiber with Small Bend Radii,” Microw. Opt. Technol. Lett. 49(9), 2133–2138 (2007).
[Crossref]

Chen, D. R.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010).
[Crossref]

Chen, X.

J. R. Zheng, P. Yan, Y. Yu, Z. Ou, J. Wang, X. Chen, and C. Du, “Temperature and index insensitive strain sensor based on a photonic crystal fiber in line Mach– Zehnder interferometer,” Opt. Commun. 297(15), 7–11 (2013).
[Crossref]

Choi, H. Y.

Choi, J. M.

Chormaic, S. N.

S. N. Chormaic, Y. Wu, and J. M. Ward, “Whispering gallery mode resonators as tools for non-linear optics and optomechanics,” Proc. SPIE 8236, 82361K (2012).
[Crossref]

A. Watkins, J. Ward, Y. Wu, and S. N. Chormaic, “Single-input spherical microbubble resonator,” Opt. Lett. 36(11), 2113–2115 (2011).
[Crossref] [PubMed]

Ciminelli, C.

Ctyroky, J.

J. Ctyroky, I. Richter, and M. Sinor, “Dual resonance in a waveguide coupled ring micro-resonator,” Opt. Quantum Electron. 38(9–11), 781–797 (2006).

De Leonardis, F.

C. E. Campanella, C. M. Campanella, F. De Leonardis, and V. M. N. Passaro, “A high efficiency label-free photonic biosensor based on vertically stacked ring resonators,” Eur. Phys. J. Spec. Top. 223, 1–13 (2014).

De Natale, P.

Dell’Olio, F.

Dong, B.

Du, C.

J. R. Zheng, P. Yan, Y. Yu, Z. Ou, J. Wang, X. Chen, and C. Du, “Temperature and index insensitive strain sensor based on a photonic crystal fiber in line Mach– Zehnder interferometer,” Opt. Commun. 297(15), 7–11 (2013).
[Crossref]

Erdogan, T.

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Fan, D.

X. Bai, D. Fan, S. Wang, S. Pu, and X. Zeng, “Strain Sensor Based on Fiber Ring Cavity Laser With Photonic Crystal Fiber In-Line Mach–Zehnder Interferometer,” IEEE Photon. J. 6(4), 6801608 (2014).
[Crossref]

Fan, X.

M. Li, X. Wu, L. Liu, X. Fan, and L. Xu, “Self-Referencing Optofluidic Ring Resonator Sensor for Highly Sensitive Biomolecular Detection,” Anal. Chem. 85(19), 9328–9332 (2013).
[Crossref] [PubMed]

Farrell, G.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of Macrobending Losses of Standard Single Mode Fiber with Small Bend Radii,” Microw. Opt. Technol. Lett. 49(9), 2133–2138 (2007).
[Crossref]

Fernández, I. M.

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

Figotin, A.

A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3), 036619 (2005).
[Crossref] [PubMed]

Freir, T.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of Macrobending Losses of Standard Single Mode Fiber with Small Bend Radii,” Microw. Opt. Technol. Lett. 49(9), 2133–2138 (2007).
[Crossref]

Gagliardi, G.

Giorgini, A.

He, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010).
[Crossref]

Huang, Y.

W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
[Crossref] [PubMed]

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Kim, M. J.

Lee, B. H.

Lee, R. K.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

J. M. Choi, R. K. Lee, and A. Yariv, “Ring fiber resonators based on fused-fiber grating add-drop filters:application to resonator coupling,” Opt. Lett. 27(18), 1598–1600 (2002).
[Crossref] [PubMed]

Li, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010).
[Crossref]

Li, M.

M. Li, X. Wu, L. Liu, X. Fan, and L. Xu, “Self-Referencing Optofluidic Ring Resonator Sensor for Highly Sensitive Biomolecular Detection,” Anal. Chem. 85(19), 9328–9332 (2013).
[Crossref] [PubMed]

Liang, W.

W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
[Crossref] [PubMed]

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Liu, L.

M. Li, X. Wu, L. Liu, X. Fan, and L. Xu, “Self-Referencing Optofluidic Ring Resonator Sensor for Highly Sensitive Biomolecular Detection,” Anal. Chem. 85(19), 9328–9332 (2013).
[Crossref] [PubMed]

Liu, Z. Y.

K. K. Qureshi, Z. Y. Liu, H. Y. Tam, and M. F. Zia, “A strain sensor based on in-line fiber Mach–Zehnder interferometer in twin-core photonic crystal fiber,” Opt. Commun. 309(15), 68–70 (2013).
[Crossref]

Malara, P.

Melloni, A.

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

Moñux, A. O.

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

Morichetti, F.

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

Ou, Z.

J. R. Zheng, P. Yan, Y. Yu, Z. Ou, J. Wang, X. Chen, and C. Du, “Temperature and index insensitive strain sensor based on a photonic crystal fiber in line Mach– Zehnder interferometer,” Opt. Commun. 297(15), 7–11 (2013).
[Crossref]

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010).
[Crossref]

Passaro, V. M. N.

C. E. Campanella, C. M. Campanella, F. De Leonardis, and V. M. N. Passaro, “A high efficiency label-free photonic biosensor based on vertically stacked ring resonators,” Eur. Phys. J. Spec. Top. 223, 1–13 (2014).

Poon, J. K. S.

Pu, S.

X. Bai, D. Fan, S. Wang, S. Pu, and X. Zeng, “Strain Sensor Based on Fiber Ring Cavity Laser With Photonic Crystal Fiber In-Line Mach–Zehnder Interferometer,” IEEE Photon. J. 6(4), 6801608 (2014).
[Crossref]

Qureshi, K. K.

K. K. Qureshi, Z. Y. Liu, H. Y. Tam, and M. F. Zia, “A strain sensor based on in-line fiber Mach–Zehnder interferometer in twin-core photonic crystal fiber,” Opt. Commun. 309(15), 68–70 (2013).
[Crossref]

Ramos, C. A.

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

Richter, I.

J. Ctyroky, I. Richter, and M. Sinor, “Dual resonance in a waveguide coupled ring micro-resonator,” Opt. Quantum Electron. 38(9–11), 781–797 (2006).

Sinor, M.

J. Ctyroky, I. Richter, and M. Sinor, “Dual resonance in a waveguide coupled ring micro-resonator,” Opt. Quantum Electron. 38(9–11), 781–797 (2006).

Strain, M. J.

C. A. Ramos, F. Morichetti, A. O. Moñux, I. M. Fernández, M. J. Strain, and A. Melloni, “Dual-Mode Coupled-Resonator Integrated Optical Filters,” IEEE Photon. Technol. Lett. 26(9), 929–932 (2014).
[Crossref]

Tam, H. Y.

K. K. Qureshi, Z. Y. Liu, H. Y. Tam, and M. F. Zia, “A strain sensor based on in-line fiber Mach–Zehnder interferometer in twin-core photonic crystal fiber,” Opt. Commun. 309(15), 68–70 (2013).
[Crossref]

Vahala, K. J.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3), 036619 (2005).
[Crossref] [PubMed]

Wang, J.

J. R. Zheng, P. Yan, Y. Yu, Z. Ou, J. Wang, X. Chen, and C. Du, “Temperature and index insensitive strain sensor based on a photonic crystal fiber in line Mach– Zehnder interferometer,” Opt. Commun. 297(15), 7–11 (2013).
[Crossref]

Wang, P.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of Macrobending Losses of Standard Single Mode Fiber with Small Bend Radii,” Microw. Opt. Technol. Lett. 49(9), 2133–2138 (2007).
[Crossref]

Wang, Q.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of Macrobending Losses of Standard Single Mode Fiber with Small Bend Radii,” Microw. Opt. Technol. Lett. 49(9), 2133–2138 (2007).
[Crossref]

Wang, S.

X. Bai, D. Fan, S. Wang, S. Pu, and X. Zeng, “Strain Sensor Based on Fiber Ring Cavity Laser With Photonic Crystal Fiber In-Line Mach–Zehnder Interferometer,” IEEE Photon. J. 6(4), 6801608 (2014).
[Crossref]

Ward, J.

Ward, J. M.

S. N. Chormaic, Y. Wu, and J. M. Ward, “Whispering gallery mode resonators as tools for non-linear optics and optomechanics,” Proc. SPIE 8236, 82361K (2012).
[Crossref]

Watkins, A.

Wei, L.

Wu, X.

M. Li, X. Wu, L. Liu, X. Fan, and L. Xu, “Self-Referencing Optofluidic Ring Resonator Sensor for Highly Sensitive Biomolecular Detection,” Anal. Chem. 85(19), 9328–9332 (2013).
[Crossref] [PubMed]

Wu, Y.

S. N. Chormaic, Y. Wu, and J. M. Ward, “Whispering gallery mode resonators as tools for non-linear optics and optomechanics,” Proc. SPIE 8236, 82361K (2012).
[Crossref]

A. Watkins, J. Ward, Y. Wu, and S. N. Chormaic, “Single-input spherical microbubble resonator,” Opt. Lett. 36(11), 2113–2115 (2011).
[Crossref] [PubMed]

Xiao, Y. F.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010).
[Crossref]

Xu, L.

M. Li, X. Wu, L. Liu, X. Fan, and L. Xu, “Self-Referencing Optofluidic Ring Resonator Sensor for Highly Sensitive Biomolecular Detection,” Anal. Chem. 85(19), 9328–9332 (2013).
[Crossref] [PubMed]

Xu, Y.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Yan, P.

J. R. Zheng, P. Yan, Y. Yu, Z. Ou, J. Wang, X. Chen, and C. Du, “Temperature and index insensitive strain sensor based on a photonic crystal fiber in line Mach– Zehnder interferometer,” Opt. Commun. 297(15), 7–11 (2013).
[Crossref]

Yang, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010).
[Crossref]

W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
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Figures (9)

Fig. 1
Fig. 1 Splitting mode resonant sensors, excited by an electric field Ei introduced by the optical coupler 1. Their transmission response is evaluated at the fiber coupler 2. (a) Fiber Bragg Grating Ring Resonator (FBGRR), where an environmental perturbation (χ) is applied to the FBG; (b) π-shifted Fiber Bragg Grating Ring Resonator (π-FBGRR), where an environmental perturbation (χ) is applied to the π-FBG region.
Fig. 2
Fig. 2 FBGRR transmission TFBGRR contour curve in the λ-range [1.560470 μm; 1.560620 μm] evaluated with li = 16 cm by changing χ from −5 × 10−5 to 5 × 10−5 with a resolution of 5 × 10−7; (Inset on the bottom) Zoom of TFBGRR in the λ-range [1.560470 μm; 1.560480 μm] being χ in the range [1 × 10−6; 5 × 10−6]; the black dashed line refers to χ = 2 × 10−6 for the two peak positions (black dots). (Insets on the right) FBG reflectivity in the same wavelength ranges.
Fig. 3
Fig. 3 π-FBGRR transmission (Tπ-FBGRR) contour curve in the λ-range [1.560470 μm; 1.560620 μm] evaluated with li = 16 cm by changing χ from −5 × 10−5 to 5 × 10−5 with a resolution of 5 × 10−7; (Inset on the bottom) Zoom of Tπ-FBGRR in the λ-range [1.560496 μm; 1.560508 μm] with χ in the range [1 × 10−6; 4 × 10−6]. The black dashed line refers to χ = 1.5 × 10−6, giving the four peak positions (black dots). (Insets on the right) π-FBG reflectivity in the same wavelength ranges.
Fig. 4
Fig. 4 π-FBGRR transmission (Tπ-FBGRR) contour curve in the λ-range [1.560499 μm; 1.560504 μm] evaluated with li = 28 cm for χ ranging from −5 × 10−6 to 5 × 10−6 with a resolution of 5 × 10−7. In the zoom (inset), the sym resonance branch is flat by imposing the condition of Eq. (18).
Fig. 5
Fig. 5 Tπ-FBGRR in the λ-range [1.5604980 μm; 1.5605019 μm] with li = 4.2 m. Contour curve by changing χ from −5 × 10−6 to 5 × 10−6 with a resolution of 5 × 10−7.
Fig. 6
Fig. 6 Splitting dynamics of a π-FBGRR splitting mode sensor: (a) Revolved splitting dynamics given by Eq. (17); (b) Flat splitting dynamics as in Eq. (18); (c) Normal cross splitting dynamics of Eq. (19).
Fig. 7
Fig. 7 Splitting magnitude of FBGRR (SplitFBGRR, blue curve) due to an applied strain ranging from 2.57 to 4 με when li = 16 cm and |Δn| = 4 × 10−5. Strain sensitivity SFBGRR (red curve).
Fig. 8
Fig. 8 Splitting magnitude of π-FBGRR (Splitπ-FBGRR, solid blue curve) and FBGRR (Splitπ-FBGRR, dashed blue curve) due to an applied strain ranging from 0 to 0.35 με when li = 28 cm and |Δn| = 4 × 10−5 (optimal case). Strain sensitivity of π-FBGRR, Sπ-FBGRR (solid red curve), and FBGRR, SFBGRR (dashed red curve).
Fig. 9
Fig. 9 Splitting magnitude (Splitπ-FBGRR) of π-FBGRR versus the applied strain for power coupling coefficients of 2% (τ = 0.99), 3% (τ = 0.985), 5% (τ = 0.975), 7% (τ = 0.965) and 10% (τ = 0.945),when li = 28 cm and |Δn| = 4 × 10−5 .

Tables (1)

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Table 1 FBG physical parameters

Equations (19)

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[ E a E b * ] = T F B G [ e j Φ 0 0 e j Φ ] T F B G [ E a * E b ] = [ A 11 A 12 A 21 A 22 ] [ E a * E b ]
T F B G = [ t r 2 / t r / t r / t 1 / t ]
t = Θ Θ cos h( Θ )+ j Δ β sin h( Θ ) r = j K sin h( Θ L ) Θ cos h( Θ ) + j Δ β sin h( Θ )
Θ = [ | K | 2 ( Δ β ) 2 ] 1 2 ; K = π | Δ n | λ B ; Δ β = 2 π n ( λ B λ λ λ B ) ;
S c = 1 A 22 [ det ( A ) A 21 A 21 det ( A ) ] = [ t π F B G r π F B G r π F B G t π F B G ]
T π F B G R R = | E o u t E i n | 2 = | 1 2 [ k 2 a e j β l i / 2 ( t π F B G + r π F B G ) 1 - τ 2 a 2 e j β l i ( t π F B G + r π F B G ) + k 2 a e j β l i / 2 ( t π F B G r π F B G ) 1 - τ 2 a 2 e j β l i ( t π F B G r π F B G ) ] | 2
T F B G R R = T π F B G R R | Φ = 0
f S ( λ ) = 1 - τ 2 a 2 e j 2 π n λ l i [ t π F B G ( λ ) + r π F B G ( λ ) ]
e j 2 π n λ R S l i = τ 2 a 2 [ t π F B G ( λ R S ) + r π F B G ( λ R S ) ] e j 2 π q
f A ( λ ) = 1 - τ 2 a 2 e j 2 π n λ l i [ t π F B G ( λ ) r π F B G ( λ ) ]
e j 2 π n λ R A l i = τ 2 a 2 [ t π F B G ( λ R A ) r π F B G ( λ R A ) ] e j 2 π q
S p l i t = | λ R S λ R A | | q = q *
Δ λ B = 2 n Δ Λ + 2 Λ Δ n g = χ λ B
e j 2 π n λ R S χ l i = τ 2 a 2 [ t π F B G ( ( 1 + χ ) λ R S ) + r π F B G ( ( 1 + χ ) λ R S ) ] e j 2 π q *
e j 2 π n λ R A χ l i = τ 2 a 2 [ t π F B G ( ( 1 + χ ) λ R A ) r π F B G ( ( 1 + χ ) λ R A ) ] e j 2 π q *
S p l i t χ = | λ R S χ λ R A χ | | q = q *
2 S p l i t χ M A X = 2 | λ R S χ λ R A χ | M A X | q = q * > F W H M π F B G
2 S p l i t χ * , l i M A X = 2 | λ R S χ * , l i λ R A χ * , l i | M A X | q = q * = F W H M π F B G
2 S p l i t χ * , l i M A X = 2 | λ R S χ * , l i λ R A χ * , l i | M A X | q = q * < < F W H M π F B G

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