Abstract

A dual-frequency optoelectronic oscillator (OEO) incorporating a polarization-maintaining fiber Bragg grating (PMFBG) Fabry-Perot filter for high-sensitivity and high-speed axial strain and temperature sensing is proposed and experimentally demonstrated. In the OEO loop, two oscillation frequencies are determined by a PMFBG Fabry-Perot filter with two ultra-narrow notches and two laser sources which operate as a dual-passband microwave photonic filter. The fiber birefringence affected by axial strain is far less than the temperature. Through monitoring the variations of two oscillating frequencies and beat frequency, the simultaneous measurement for the axial strain and temperature is realized. The sensitivities of the proposed OEO sensor for axial strain and temperature are experimentally measured to be as high as 100.6 or 100.5 MHz/με and −41 MHz/°C, respectively.

© 2017 Optical Society of America

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References

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  1. S. M. Lee, S. S. Saini, and M. Y. Jeong, “Simultaneous Measurement of Refractive Index, Temperature, and Strain Using Etched-Core Fiber Bragg Grating Sensors,” IEEE Photonics Technol. Lett. 22(19), 1431–1433 (2010).
    [Crossref]
  2. W. Yiping, M. Wang, and X. Huang, “In fiber Bragg grating twist sensor based on analysis of polarization dependent loss,” Opt. Express 21(10), 11913–11920 (2013).
    [Crossref] [PubMed]
  3. Y. Dai, M. Yang, G. Xu, and Y. Yuan, “Magnetic field sensor based on fiber Bragg grating with a spiral microgroove ablated by femtosecond laser,” Opt. Express 21(14), 17386–17391 (2013).
    [Crossref] [PubMed]
  4. B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
    [Crossref]
  5. Y. Ran, Y.-N. Tan, L.-P. Sun, S. Gao, J. Li, L. Jin, and B.-O. Guan, “193 nm excimer laser inscribed Bragg gratings in microfibers for refractive index sensing,” Opt. Express 19(19), 18577–18583 (2011).
    [Crossref] [PubMed]
  6. J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
    [Crossref]
  7. Y. P. Wang, X. Q. Huang, and M. Wang, “Temperature- and strain-independent torsion sensor utilising pol arisation-dependent loss of Hi-Bi FBGs,” Electron. Lett. 49(13), 840–841 (2013).
    [Crossref]
  8. S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. (accepted).
  9. J. P. Yao, “Microwave photonics for high resolution and high speed interrogation of fiber Bragg grating sensors,” Fiber Integr. Opt. 34(4), 230–242 (2015).
    [Crossref]
  10. J. P. Yao, “Optoelectronic oscillator for high speed and high resolution optical sensing,” J. Lightwave Technol. (accepted).
  11. M. Li, W. Z. Li, J. P. Yao, and J. Azana, “Femtometer-resolution wavelength interrogation using an optoelectronic oscillator,” in IPC 2012 (2012).
  12. F. Kong, W. Li, and J. Yao, “Transverse load sensing based on a dual-frequency optoelectronic oscillator,” Opt. Lett. 38(14), 2611–2613 (2013).
    [Crossref] [PubMed]
  13. O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
    [Crossref]
  14. X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
    [Crossref]
  15. Y. P. Wang, J. J. Zhang, and J. P. Yao, “An optoelectronic oscillator for high sensitivity temperature sensing,” IEEE Photonics Technol. Lett. 28(13), 1458–1461 (2016).
    [Crossref]
  16. X. Zou, M. Li, W. Pan, B. Luo, L. Yan, and L. Shao, “Optical length change measurement via RF frequency shift analysis of incoherent light source based optoelectronic oscillator,” Opt. Express 22(9), 11129–11139 (2014).
    [Crossref] [PubMed]
  17. B.-O. Guan, L. Jin, Y. Zhang, and H.-Y. Tam, “Polarimetric heterodyning fiber grating laser sensors,” J. Lightwave Technol. 30(8), 1097–1112 (2012).
    [Crossref]
  18. G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
    [Crossref]

2017 (1)

O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
[Crossref]

2016 (2)

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

Y. P. Wang, J. J. Zhang, and J. P. Yao, “An optoelectronic oscillator for high sensitivity temperature sensing,” IEEE Photonics Technol. Lett. 28(13), 1458–1461 (2016).
[Crossref]

2015 (1)

J. P. Yao, “Microwave photonics for high resolution and high speed interrogation of fiber Bragg grating sensors,” Fiber Integr. Opt. 34(4), 230–242 (2015).
[Crossref]

2014 (2)

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

X. Zou, M. Li, W. Pan, B. Luo, L. Yan, and L. Shao, “Optical length change measurement via RF frequency shift analysis of incoherent light source based optoelectronic oscillator,” Opt. Express 22(9), 11129–11139 (2014).
[Crossref] [PubMed]

2013 (5)

2012 (1)

2011 (1)

2010 (1)

S. M. Lee, S. S. Saini, and M. Y. Jeong, “Simultaneous Measurement of Refractive Index, Temperature, and Strain Using Etched-Core Fiber Bragg Grating Sensors,” IEEE Photonics Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

2004 (1)

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Albert, J.

J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Bai, Y. L.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Caucheteur, C.

J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Chen, G. H.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Dai, Y.

Deng, H.

O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
[Crossref]

Feng, S. C.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Gao, S.

Guan, B.-O.

Huang, X.

Huang, X. Q.

Y. P. Wang, X. Q. Huang, and M. Wang, “Temperature- and strain-independent torsion sensor utilising pol arisation-dependent loss of Hi-Bi FBGs,” Electron. Lett. 49(13), 840–841 (2013).
[Crossref]

Jeong, M. Y.

S. M. Lee, S. S. Saini, and M. Y. Jeong, “Simultaneous Measurement of Refractive Index, Temperature, and Strain Using Etched-Core Fiber Bragg Grating Sensors,” IEEE Photonics Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

Jia, H. Z.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Jian, S. S.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Jin, L.

Kong, F.

Lee, S. M.

S. M. Lee, S. S. Saini, and M. Y. Jeong, “Simultaneous Measurement of Refractive Index, Temperature, and Strain Using Etched-Core Fiber Bragg Grating Sensors,” IEEE Photonics Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

Li, H. S.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Li, J.

Li, M.

Li, P. X.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

Li, W.

Li, W. Z.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

Liu, L. Y.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Liu, S.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Liu, X. K.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

Liu, Z. B.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Luo, B.

Pan, S. L.

S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. (accepted).

Pan, W.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

X. Zou, M. Li, W. Pan, B. Luo, L. Yan, and L. Shao, “Optical length change measurement via RF frequency shift analysis of incoherent light source based optoelectronic oscillator,” Opt. Express 22(9), 11129–11139 (2014).
[Crossref] [PubMed]

Peng, W. J.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Ran, Y.

Saini, S. S.

S. M. Lee, S. S. Saini, and M. Y. Jeong, “Simultaneous Measurement of Refractive Index, Temperature, and Strain Using Etched-Core Fiber Bragg Grating Sensors,” IEEE Photonics Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

Shao, L.

Shao, L. Y.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Sun, L.-P.

Tam, H.-Y.

Tan, Y.-N.

Wang, M.

Y. P. Wang, X. Q. Huang, and M. Wang, “Temperature- and strain-independent torsion sensor utilising pol arisation-dependent loss of Hi-Bi FBGs,” Electron. Lett. 49(13), 840–841 (2013).
[Crossref]

W. Yiping, M. Wang, and X. Huang, “In fiber Bragg grating twist sensor based on analysis of polarization dependent loss,” Opt. Express 21(10), 11913–11920 (2013).
[Crossref] [PubMed]

Wang, W. C.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Wang, Y. P.

Y. P. Wang, J. J. Zhang, and J. P. Yao, “An optoelectronic oscillator for high sensitivity temperature sensing,” IEEE Photonics Technol. Lett. 28(13), 1458–1461 (2016).
[Crossref]

Y. P. Wang, X. Q. Huang, and M. Wang, “Temperature- and strain-independent torsion sensor utilising pol arisation-dependent loss of Hi-Bi FBGs,” Electron. Lett. 49(13), 840–841 (2013).
[Crossref]

Xu, G.

Xu, L.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Xu, O.

O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
[Crossref]

Yan, L.

Yan, L. S.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

Yang, M.

Yao, J.

Yao, J. P.

O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
[Crossref]

Y. P. Wang, J. J. Zhang, and J. P. Yao, “An optoelectronic oscillator for high sensitivity temperature sensing,” IEEE Photonics Technol. Lett. 28(13), 1458–1461 (2016).
[Crossref]

J. P. Yao, “Microwave photonics for high resolution and high speed interrogation of fiber Bragg grating sensors,” Fiber Integr. Opt. 34(4), 230–242 (2015).
[Crossref]

J. P. Yao, “Optoelectronic oscillator for high speed and high resolution optical sensing,” J. Lightwave Technol. (accepted).

S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. (accepted).

Yin, B.

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

Yiping, W.

Yu, J. M.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Yuan, Y.

Zhang, J. J.

O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
[Crossref]

Y. P. Wang, J. J. Zhang, and J. P. Yao, “An optoelectronic oscillator for high sensitivity temperature sensing,” IEEE Photonics Technol. Lett. 28(13), 1458–1461 (2016).
[Crossref]

Zhang, Y.

Zou, X.

Zou, X. H.

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

Electron. Lett. (1)

Y. P. Wang, X. Q. Huang, and M. Wang, “Temperature- and strain-independent torsion sensor utilising pol arisation-dependent loss of Hi-Bi FBGs,” Electron. Lett. 49(13), 840–841 (2013).
[Crossref]

Fiber Integr. Opt. (1)

J. P. Yao, “Microwave photonics for high resolution and high speed interrogation of fiber Bragg grating sensors,” Fiber Integr. Opt. 34(4), 230–242 (2015).
[Crossref]

IEEE J. Quantum Electron. (1)

X. H. Zou, X. K. Liu, W. Z. Li, P. X. Li, W. Pan, L. S. Yan, and L. Y. Shao, “Optoelectronic Oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (5)

Y. P. Wang, J. J. Zhang, and J. P. Yao, “An optoelectronic oscillator for high sensitivity temperature sensing,” IEEE Photonics Technol. Lett. 28(13), 1458–1461 (2016).
[Crossref]

O. Xu, J. J. Zhang, H. Deng, and J. P. Yao, “Dual-frequency Optoelectronic Oscillator for Thermal-Insensitive Interrogation of a FBG Strain Sensor,” IEEE Photonics Technol. Lett. 29(4), 357–360 (2017).
[Crossref]

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

S. M. Lee, S. S. Saini, and M. Y. Jeong, “Simultaneous Measurement of Refractive Index, Temperature, and Strain Using Etched-Core Fiber Bragg Grating Sensors,” IEEE Photonics Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

B. Yin, H. S. Li, S. C. Feng, Y. L. Bai, Z. B. Liu, W. J. Peng, S. Liu, and S. S. Jian, “Temperature-Independent and Strain-Independent Twist Sensor Based on Structured PM-CFBG,” IEEE Photonics Technol. Lett. 26(15), 1565–1568 (2014).
[Crossref]

J. Lightwave Technol. (1)

Laser Photonics Rev. (1)

J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Other (3)

J. P. Yao, “Optoelectronic oscillator for high speed and high resolution optical sensing,” J. Lightwave Technol. (accepted).

M. Li, W. Z. Li, J. P. Yao, and J. Azana, “Femtometer-resolution wavelength interrogation using an optoelectronic oscillator,” in IPC 2012 (2012).

S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. (accepted).

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Figures (7)
Fig. 1
Fig. 1 Configuration of the dual-frequency OEO based on PMFBG Fabry-Perot filter for simultaneous axial strain and temperature sensing.

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Fig. 2
Fig. 2 Dual-frequency oscillating principle and sensing principle of the proposed OEO for measuring axial strain and temperature.

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Fig. 3
Fig. 3 Measured transmission spectra of the PMFBG Fabry-Perot filter at X and Y polarization directions respectively.

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Fig. 4
Fig. 4 Experimental measured frequency spectra of the generated microwave signals when different axial strain is applied to PMFBG Fabry-Perot filter at constant room temperature.

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Fig. 5
Fig. 5 (a) Relationship between the applied axial strain and the frequencies of two microwave signals; (b) Relationship between the applied axial strain and the frequency of the beat signal.

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Fig. 6
Fig. 6 Experimental measured frequency spectra of the generated microwave signals when different temperatures are applied to PMFBG Fabry-Perot filter without axial strain.

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Fig. 7
Fig. 7 (a) Relationship between the temperature and the frequencies of two microwave signals; (b) Relationship between the temperature and the frequency of the beat signal.

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Equations (6)

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

f X , Y o s c = | f X , Y − f s | ≈ c n 0 ( | λ s − λ X , Y | λ s 2 )
Δ f = | f X o s c − f Y o s c | = c B / n 0 λ 0
{ Δ n X = − ( n X 3 / 2 ) [ p 12 − ν ( p 11 + p 12 ) ] ε Δ n Y = − ( n Y 3 / 2 ) [ p 12 − ν ( p 11 + p 12 ) ] ε
B = α T × G × E × C × ( T S − T ) / [ 2 × ( 1 − ν ) ]
{ Δ f X o s c = K X T Δ T + K ε Δ ε Δ f Y o s c = K Y T Δ T + K ε Δ ε Δ f = | K Y T − K Y T | Δ T
{ Δ f X o s c = 1152 Δ T + 101 Δ ε Δ f Y o s c = 1193 Δ T + 101 Δ ε Δ f = − 41 Δ T

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