Abstract

We report a large-scale multi-channel fiber sensing network, where ultra-short FBGs (USFBGs) instead of conventional narrow-band ultra-weak FBGs are used as the sensors. In the time division multiplexing scheme of the network, each grating response is resolved as three adjacent discrete peaks. The central wavelengths of USFBGs are tracked with the differential detection, which is achieved by calculating the peak-to-peak ratio of two maximum peaks. Compared with previous large-scale hybrid multiplexing sensing networks (e.g., WDM/TDM) which typically have relatively low interrogation speed and very high complexity, the proposed system can achieve interrogation of all channel sensors through very fast and simple intensity measurements with a broad dynamic range. A proof-of-concept experiment with twenty USFBGs, at two wavelength channels, was performed and a fast static strain measurements were demonstrated, with a high average sensitivity of ~0.54dB/µƐ and wide dynamic range of over ~3000µƐ. The channel to channel switching time was 10ms and total network interrogation time was 50ms.

© 2017 Optical Society of America

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References

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  1. H. Guo, F. Liu, Y. Yuan, H. Yu, and M. Yang, “Ultra-weak FBG and its refractive index distribution in the drawing optical fiber,” Opt. Express 23(4), 4829–4838 (2015).
    [Crossref] [PubMed]
  2. X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
    [Crossref]
  3. Z. Luo, H. Wen, H. Guo, and M. Yang, “A time- and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings,” Opt. Express 21(19), 22799–22807 (2013).
    [Crossref] [PubMed]
  4. A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
    [Crossref]
  5. W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
    [Crossref]
  6. P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
    [Crossref]
  7. Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
    [Crossref]
  8. L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
    [Crossref]
  9. Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.
  10. J. Rohollahnejad, L. Xia, and R. Cheng, “Shifted Optical Gaussian Filters based Time Division Multiplexing of USFBGs Sensing Network,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTu5A.91.
    [Crossref]
  11. W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
    [Crossref]
  12. C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
    [Crossref]
  13. P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
    [Crossref]

2015 (2)

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

H. Guo, F. Liu, Y. Yuan, H. Yu, and M. Yang, “Ultra-weak FBG and its refractive index distribution in the drawing optical fiber,” Opt. Express 23(4), 4829–4838 (2015).
[Crossref] [PubMed]

2013 (2)

Z. Luo, H. Wen, H. Guo, and M. Yang, “A time- and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings,” Opt. Express 21(19), 22799–22807 (2013).
[Crossref] [PubMed]

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

2012 (2)

2008 (1)

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

2003 (1)

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

2001 (1)

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

2000 (2)

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Cerecedo-Núñez, H. H.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Chan, C. C.

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

Chan, P. K.

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

Chen, P. K. C.

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Delbeke, D.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Demokan, M. S.

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

Dong, B.

W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Dumon, P.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Gong, J.

Guo, H.

Jin, W.

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Li, X.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Liu, B.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Liu, D.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Liu, F.

Liu, H.

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Liu, Q.

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Luo, B.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Luo, Z.

Miao, Y.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Pan, W.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Pathak, S.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Pickrell, G.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Qi, B.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Shillig, T. J.

Sun, Q.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Trita, A.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Van Thourhout, D.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Vermeiren, J.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Voet, E.

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

Wang, A.

W. Wang, J. Gong, B. Dong, D. Y. Wang, T. J. Shillig, and A. Wang, “A large serial time-division multiplexed fiber bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Wang, D. Y.

Wang, P.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Wang, W.

Wen, H.

Wo, J.

Wu, Z.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Xu, H. G.

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Yan, L.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Yang, M.

Yu, H.

Yuan, Y.

Zhang, M.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid tdm/wdm-based fiber-optic sensor network for perimeter intrusion detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).
[Crossref]

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

Zhang, P.

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Zhang, W.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Zhang, Z.

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

Zhao, Q.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Zhou, H.

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

Zhou, Y.

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

P. K. Chan, W. Jin, and M. S. Demokan, “Fmcw multiplexing of fiber bragg grating sensors,” IEEE J. Sel. Top. Quantum Electron. 6(5), 756–763 (2000).
[Crossref]

IEEE Photonics J. (2)

A. Trita, E. Voet, J. Vermeiren, D. Delbeke, P. Dumon, S. Pathak, and D. Van Thourhout, “Simultaneous interrogation of multiple fiber bragg grating sensors using an arrayed waveguide grating filter fabricated in soi platform,” IEEE Photonics J. 7(6), 1–11 (2015).
[Crossref]

L. Yan, Z. Wu, Z. Zhang, W. Pan, B. Luo, and P. Wang, “High-speed fbg-based fiber sensor networks for semidistributed strain measurements,” IEEE Photonics J. 5(2), 7200507 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. Miao, B. Liu, W. Zhang, B. Dong, H. Zhou, and Q. Zhao, “Dynamic temperature compensating interrogation technique for strain sensors with tilted fiber bragg gratings,” IEEE Photonics Technol. Lett. 20(16), 1393–1395 (2008).
[Crossref]

J. Lightwave Technol. (2)

Opt. Eng. (1)

P. Zhang, H. H. Cerecedo-Núñez, B. Qi, G. Pickrell, and A. Wang, “Optical time-domain reflectometry interrogation of multiplexing low reflectance Bragg-grating-based sensor system,” Opt. Eng. 42(6), 1597–1603 (2003).
[Crossref]

Opt. Express (2)

Proc. SPIE (1)

C. C. Chan, W. Jin, and M. S. Demokan, “TDM of FBG sensors by use of a tunable laser source,” Proc. SPIE 4357, 77–86 (2001).
[Crossref]

Sens. Actuators A Phys. (1)

W. Jin, Y. Zhou, P. K. C. Chen, and H. G. Xu, “A fiber-optic grating sensor for the study of flow-induced vibrations,” Sens. Actuators A Phys. 79(1), 36–45 (2000).
[Crossref]

Other (2)

Q. Sun, X. Li, M. Zhang, Q. Liu, H. Liu, and D. Liu, “High capacity fiber optic sensor networks using hybrid multiplexing techniques and their applications,” in International Conference on Optical Instruments and Technology (OIT, 2013), pp. 90440.

J. Rohollahnejad, L. Xia, and R. Cheng, “Shifted Optical Gaussian Filters based Time Division Multiplexing of USFBGs Sensing Network,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTu5A.91.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 USFBG spectrum and the three laser lines.
Fig. 2
Fig. 2 Sensing sensitivity and measurement range for different Bragg grating bandwidths.
Fig. 3
Fig. 3 Basic structure of the proposed sensing setup. OMBFP: optical multi-bandpass filter, EOM: electro-optic modulator, PC: polarization controller, PD: photodetector.
Fig. 4
Fig. 4 OMBPF three port periodic transmission function.
Fig. 5
Fig. 5 Identical USFBGs spectrum and wavelength lines of laser for two channels.
Fig. 6
Fig. 6 First channel time domain signals of three APDs under (a) zero strain, (b) ~500 µƐ, (c) ~750 µƐ, (d) ~1500 µƐ, (e) ~2250 µƐ and (f) ~3000 µƐ, (APD1, APD2 and ADP3 signals are shown in blue, red and green respectively).
Fig. 7
Fig. 7 (a) The evolution of the USFBG12 spectrum as the applied strain increases, (b) three APDs peak responses under different strains and (c) differential values [log(I2)-log(I1)] and [log(I2)-log(I3)] vs. strain, red and blue line, respectively.
Fig. 8
Fig. 8 Multi-reflection crosstalk level versus grating bandwidth
Fig. 9
Fig. 9 (a) Spectrum distortion of the 1000th grating due to spectral shadowing effect. (b) Response curve change of the 1000th grating.

Equations (7)

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

I= 0 R( λ λ B ) S( λ λ l )dλ
I=C S λ l R m exp[ ( 4ln2 ) ( λ l λ B Δ λ B ) 2 ]
I 1 = C 1 S λ 1 R m exp[ ( 4ln2 ) ( ( λ 1 λ B ) / Δ λ B ) 2 ] I 2 = C 2 S λ 1 R m exp[ ( 4ln2 ) ( ( λ 1 +Δλ λ B ) / Δ λ B ) 2 ] I 3 = C 3 S λ 1 R m exp[ ( 4ln2 ) ( ( λ 1 +2Δλ λ B ) / Δ λ B ) 2 ]
{ P( λ B )=log( I 2 ( λ B ) I 1 ( λ B ) ) if( I 1 I 3 ) P( λ B )=log( I 2 ( λ B ) I 3 ( λ B ) ) if( I 1 < I 3 ) ( E+F λ B )
ΔP( λ B )=log( ( I 2 ( λ B )+Δ I 2 ) / ( I 1 ( λ B )+Δ I 1 ) )
I k (λ)= λ ( 1R( λ λ B ) ) 2k2 R( λ λ B )S( λ λ l )dλ = i=0 2(k1) C 2(i1) i (1) i R m i+1 λ R( λ λ B ) S( λ λ l )dλ
I k ( λ l ) R m S λ l exp[ ( 4ln2 ) ( λ l λ B Δ λ B ) 2 ] i=0 2(k1) C 2(k1) i (1) i R m i+1 / i+1

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