Abstract

An optical stress sensor is proposed by using a single crystal with both electro-optic and photoelastic effects. Different from previous crystal-based stress sensors, the proposed sensor is based on electro-optic compensation for stress-induced birefringence and does not need an additional quarter-wave plate or modulator, because the stress-sensing element is simultaneously used as an electro-optic compensator. Candidate sensing materials include electro-optic crystals of the 3m symmetry group and all glass with large Kerr coefficients. A primary experiment has demonstrated that the stress-induced birefringence in lithium niobate crystal can be compensated by its electro-optic birefringence. The proposed stress sensor is compact and low cost, and it is possible to achieve closed-loop stress measurement.

© 2011 Optical Society of America

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  1. S. Tai, K. Kyuma, and M. Nunoshita, “Fiber-optic acceleration sensor based on the photoelastic effect,” Appl. Opt. 22, 1771–1774 (1983).
    [CrossRef] [PubMed]
  2. A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
    [CrossRef]
  3. D. L. Tang, Z. Liang, X. D. Zhang, S. He, and F. Guo, “Three-component hybrid-integrated optical accelerometer based on LiNbO3 photoelastic waveguide,” Chin. Opt. Lett. 7, 32–35(2009).
    [CrossRef]
  4. H. Y. Fu, C. Wu, M. L. V. Tse, L. Zhang, K. C. Cheng, H. Y. Tam, B. Guan, and C. Lu, “High pressure sensor based on photonics crystal fiber for downhole application,” Appl. Opt. 49, 2639–2643 (2010).
    [CrossRef]
  5. M. Pang, H. F. Xuan, J. Ju, and W. Jin, “Influence of strain and pressure to the effective refractive index of the fundamental mode of hollow-core photonic bandgap fibers,” Opt. Express 18, 14041–14055 (2010).
    [CrossRef] [PubMed]
  6. A. K. Bhowmik, “On photoelastic stress measurements in optically absorbing medium,” Opt. Commun. 210, 165–172(2002).
    [CrossRef]
  7. W. Holzapfel and W. Settgast, “Force to frequency conversion by intracavity photoelastic modulation,” Appl. Opt. 28, 4585–4594 (1989).
    [CrossRef] [PubMed]
  8. C. Li, “Stepped polarization states: representation and its applications to optical sensing and measurement,” Opt. Commun. 281, 2033–2039 (2008).
    [CrossRef]
  9. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag1979).
  10. M. R. Hutsel, R. R. Ingle, and T. K. Gaylord, “Technique and apparatus for accurate cross-sectional stress profiling of optical fibers,” IEEE Trans. Instrum. Meas. 60, 971–979(2011).
    [CrossRef]
  11. J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
    [CrossRef]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  13. I. P. Kaminow, “Strain effects in electrooptic light modulators,” Appl. Opt. 3, 511–515 (1964).
    [CrossRef]
  14. T. J. Wang and J. S. Chung, “Electro-optically wavelength-tunable polarization converter utilizing strain-optic effect on X-cut LiNbO3,” IEEE Photon. Technol. Lett. 16, 2275–2277(2004).
    [CrossRef]
  15. D. Psaltis, H. Lee, and G. Sirat, “Acousto-electro-optic light modulation,” Appl. Phys. Lett. 46, 215–217 (1985).
    [CrossRef]
  16. R. Petkovšek, F. Bammer, D. Schuöcker, and J. Možina, “Dual-mode single-crystal photoelastic modulator and possible applications,” Appl. Opt. 48, C86–C91 (2009).
    [CrossRef] [PubMed]
  17. A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
    [CrossRef]
  18. K. S. Lee, “New compensation method for bulk optical sensors with multiple birefringences,” Appl. Opt. 28, 2001–2011(1989).
    [CrossRef] [PubMed]
  19. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, 2nd ed. (Clarendon, 1985).
  20. S. M. Hauser, L. S. Smith, D. G. Marlowe, and P. R. Yoder, “The stressed-plate shutter, a new moderate-speed electro-optical light modulator,” Appl. Opt. 2, 1175–1179 (1963).
    [CrossRef]
  21. C. Li, “Proposal for electro-optic multiplier based on dual transverse electro-optic Kerr effect,” Appl. Opt. 47, 5701–5705(2008).
    [CrossRef] [PubMed]
  22. H. C. Lefevre, The Fiber-Optic Gyroscope (Artech House, 1993).
  23. G. Martens, J. Kordts, and G. Weidinger, “Loss-compensated photoelastic fiber optic pressure sensor,” Appl. Opt. 28, 5149–5152 (1989).
    [CrossRef] [PubMed]
  24. C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843–849(2002).
    [CrossRef]

2011 (2)

M. R. Hutsel, R. R. Ingle, and T. K. Gaylord, “Technique and apparatus for accurate cross-sectional stress profiling of optical fibers,” IEEE Trans. Instrum. Meas. 60, 971–979(2011).
[CrossRef]

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

2010 (2)

2009 (2)

2008 (2)

C. Li, “Stepped polarization states: representation and its applications to optical sensing and measurement,” Opt. Commun. 281, 2033–2039 (2008).
[CrossRef]

C. Li, “Proposal for electro-optic multiplier based on dual transverse electro-optic Kerr effect,” Appl. Opt. 47, 5701–5705(2008).
[CrossRef] [PubMed]

2005 (1)

A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
[CrossRef]

2004 (1)

T. J. Wang and J. S. Chung, “Electro-optically wavelength-tunable polarization converter utilizing strain-optic effect on X-cut LiNbO3,” IEEE Photon. Technol. Lett. 16, 2275–2277(2004).
[CrossRef]

2002 (2)

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843–849(2002).
[CrossRef]

A. K. Bhowmik, “On photoelastic stress measurements in optically absorbing medium,” Opt. Commun. 210, 165–172(2002).
[CrossRef]

1992 (1)

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

1989 (3)

1985 (1)

D. Psaltis, H. Lee, and G. Sirat, “Acousto-electro-optic light modulation,” Appl. Phys. Lett. 46, 215–217 (1985).
[CrossRef]

1983 (1)

1964 (1)

1963 (1)

Bammer, F.

Bhowmik, A. K.

A. K. Bhowmik, “On photoelastic stress measurements in optically absorbing medium,” Opt. Commun. 210, 165–172(2002).
[CrossRef]

Chauvet, M.

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

Cheng, K. C.

Chung, J. S.

T. J. Wang and J. S. Chung, “Electro-optically wavelength-tunable polarization converter utilizing strain-optic effect on X-cut LiNbO3,” IEEE Photon. Technol. Lett. 16, 2275–2277(2004).
[CrossRef]

Degiorgio, V.

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

Fang, X.

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

Fu, H. Y.

Gadri, S. B.

A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
[CrossRef]

Garzarella, A.

A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
[CrossRef]

Gaylord, T. K.

M. R. Hutsel, R. R. Ingle, and T. K. Gaylord, “Technique and apparatus for accurate cross-sectional stress profiling of optical fibers,” IEEE Trans. Instrum. Meas. 60, 971–979(2011).
[CrossRef]

Gdoutos, E. E.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag1979).

Guan, B.

Guo, F.

Hauser, S. M.

He, S.

D. L. Tang, Z. Liang, X. D. Zhang, S. He, and F. Guo, “Three-component hybrid-integrated optical accelerometer based on LiNbO3 photoelastic waveguide,” Chin. Opt. Lett. 7, 32–35(2009).
[CrossRef]

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

Holzapfel, W.

Hutsel, M. R.

M. R. Hutsel, R. R. Ingle, and T. K. Gaylord, “Technique and apparatus for accurate cross-sectional stress profiling of optical fibers,” IEEE Trans. Instrum. Meas. 60, 971–979(2011).
[CrossRef]

Ingle, R. R.

M. R. Hutsel, R. R. Ingle, and T. K. Gaylord, “Technique and apparatus for accurate cross-sectional stress profiling of optical fibers,” IEEE Trans. Instrum. Meas. 60, 971–979(2011).
[CrossRef]

Jin, W.

Jin, X.

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

Ju, J.

Kaminow, I. P.

Kordts, J.

Kyuma, K.

Lee, H.

D. Psaltis, H. Lee, and G. Sirat, “Acousto-electro-optic light modulation,” Appl. Phys. Lett. 46, 215–217 (1985).
[CrossRef]

Lee, K. S.

Lefevre, H. C.

H. C. Lefevre, The Fiber-Optic Gyroscope (Artech House, 1993).

Li, C.

C. Li, “Proposal for electro-optic multiplier based on dual transverse electro-optic Kerr effect,” Appl. Opt. 47, 5701–5705(2008).
[CrossRef] [PubMed]

C. Li, “Stepped polarization states: representation and its applications to optical sensing and measurement,” Opt. Commun. 281, 2033–2039 (2008).
[CrossRef]

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843–849(2002).
[CrossRef]

Liang, Z.

Lin, J.

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

Lu, C.

Marlowe, D. G.

Martens, G.

Minzioni, P.

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

Možina, J.

Nunoshita, M.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, 2nd ed. (Clarendon, 1985).

Pang, M.

Parravicini, J.

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

Petkovšek, R.

Psaltis, D.

D. Psaltis, H. Lee, and G. Sirat, “Acousto-electro-optic light modulation,” Appl. Phys. Lett. 46, 215–217 (1985).
[CrossRef]

Safioui, J.

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

Schuöcker, D.

Settgast, W.

Sirat, G.

D. Psaltis, H. Lee, and G. Sirat, “Acousto-electro-optic light modulation,” Appl. Phys. Lett. 46, 215–217 (1985).
[CrossRef]

Smith, L. S.

Tai, S.

Tam, H. Y.

Tang, D. L.

Theocaris, P. S.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag1979).

Tse, M. L. V.

Wang, A.

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

Wang, T. J.

T. J. Wang and J. S. Chung, “Electro-optically wavelength-tunable polarization converter utilizing strain-optic effect on X-cut LiNbO3,” IEEE Photon. Technol. Lett. 16, 2275–2277(2004).
[CrossRef]

Weidinger, G.

Wieting, T. J.

A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
[CrossRef]

Wu, C.

Wu, D. H.

A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
[CrossRef]

Xuan, H. F.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yoder, P. R.

Yoshino, T.

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843–849(2002).
[CrossRef]

Zhang, L.

Zhang, X. D.

Appl. Opt. (9)

S. Tai, K. Kyuma, and M. Nunoshita, “Fiber-optic acceleration sensor based on the photoelastic effect,” Appl. Opt. 22, 1771–1774 (1983).
[CrossRef] [PubMed]

H. Y. Fu, C. Wu, M. L. V. Tse, L. Zhang, K. C. Cheng, H. Y. Tam, B. Guan, and C. Lu, “High pressure sensor based on photonics crystal fiber for downhole application,” Appl. Opt. 49, 2639–2643 (2010).
[CrossRef]

W. Holzapfel and W. Settgast, “Force to frequency conversion by intracavity photoelastic modulation,” Appl. Opt. 28, 4585–4594 (1989).
[CrossRef] [PubMed]

I. P. Kaminow, “Strain effects in electrooptic light modulators,” Appl. Opt. 3, 511–515 (1964).
[CrossRef]

R. Petkovšek, F. Bammer, D. Schuöcker, and J. Možina, “Dual-mode single-crystal photoelastic modulator and possible applications,” Appl. Opt. 48, C86–C91 (2009).
[CrossRef] [PubMed]

K. S. Lee, “New compensation method for bulk optical sensors with multiple birefringences,” Appl. Opt. 28, 2001–2011(1989).
[CrossRef] [PubMed]

S. M. Hauser, L. S. Smith, D. G. Marlowe, and P. R. Yoder, “The stressed-plate shutter, a new moderate-speed electro-optical light modulator,” Appl. Opt. 2, 1175–1179 (1963).
[CrossRef]

C. Li, “Proposal for electro-optic multiplier based on dual transverse electro-optic Kerr effect,” Appl. Opt. 47, 5701–5705(2008).
[CrossRef] [PubMed]

G. Martens, J. Kordts, and G. Weidinger, “Loss-compensated photoelastic fiber optic pressure sensor,” Appl. Opt. 28, 5149–5152 (1989).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

D. Psaltis, H. Lee, and G. Sirat, “Acousto-electro-optic light modulation,” Appl. Phys. Lett. 46, 215–217 (1985).
[CrossRef]

Chin. Opt. Lett. (1)

IEEE Photon. Technol. Lett. (1)

T. J. Wang and J. S. Chung, “Electro-optically wavelength-tunable polarization converter utilizing strain-optic effect on X-cut LiNbO3,” IEEE Photon. Technol. Lett. 16, 2275–2277(2004).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

M. R. Hutsel, R. R. Ingle, and T. K. Gaylord, “Technique and apparatus for accurate cross-sectional stress profiling of optical fibers,” IEEE Trans. Instrum. Meas. 60, 971–979(2011).
[CrossRef]

J. Appl. Phys. (2)

J. Parravicini, J. Safioui, V. Degiorgio, P. Minzioni, and M. Chauvet, “All-optical technique to measure the pyroelectric coefficient in electro-optic crystals,” J. Appl. Phys. 109, 033106 (2011).
[CrossRef]

A. Garzarella, S. B. Gadri, T. J. Wieting, and D. H. Wu, “Piezo-induced sensitivity enhancements in electro-optic sensors,” J. Appl. Phys. 98, 043113 (2005).
[CrossRef]

J. Lightwave Technol. (2)

A. Wang, S. He, X. Fang, X. Jin, and J. Lin, “Optical fiber pressure sensor based on photoelasticity and its application,” J. Lightwave Technol. 10, 1466–1472 (1992).
[CrossRef]

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843–849(2002).
[CrossRef]

Opt. Commun. (2)

A. K. Bhowmik, “On photoelastic stress measurements in optically absorbing medium,” Opt. Commun. 210, 165–172(2002).
[CrossRef]

C. Li, “Stepped polarization states: representation and its applications to optical sensing and measurement,” Opt. Commun. 281, 2033–2039 (2008).
[CrossRef]

Opt. Express (1)

Other (4)

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag1979).

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, 2nd ed. (Clarendon, 1985).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

H. C. Lefevre, The Fiber-Optic Gyroscope (Artech House, 1993).

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Figures (4)

Fig. 1
Fig. 1

(a)  3 m crystal and its principal axes in the presence of applied electric field E 2 and stress σ 2 along the x 2 axis, where φ is the angle between the o x 3 and o x 3 axes; (b) cubic crystal or glass in the presence of applied electric field E 1 and stress σ 1 along the x 1 axis.

Fig. 2
Fig. 2

(a) Experimental setup for the photoelastic stress sensor using a LN crystal, where LD is a collimating diode laser, PD is a photodetector, P1 and P2 are two prism polarizers, the LN crystal and PZT transducer are clamped in a holder as shown in (b), and POF is a plastic optical fiber.

Fig. 3
Fig. 3

Schematic drawing of birefringence Δ n as a function of applied stress and electric field ( E 2 , σ 2 ), where every point on curve AD can satisfy the compensating condition of Δ n = 0 .

Fig. 4
Fig. 4

Light transmission ratio I o / I i as a function of applied stress when the compensating electric field satisfies Eq. (6).

Equations (19)

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( 1 n o 2 r 22 E 2 + π 12 σ 2 ) x 1 2 + ( 1 n o 2 + r 22 E 2 + π 22 σ 2 ) x 2 2 + ( 1 n e 2 + π 32 σ 2 ) x 3 2 + 2 ( r 51 E 2 + π 42 σ 2 ) x 2 x 3 = 1 ,
{ π 12 = p 11 s 12 + p 12 s 11 + p 13 s 13 p 14 s 14 π 22 = π 11 = p 12 s 12 + p 11 s 11 + p 13 s 13 + p 14 s 14 π 32 = π 31 = p 31 s 12 + p 31 s 11 + p 33 s 13 π 42 = π 41 = p 41 s 12 p 41 s 11 p 44 s 14 ,
{ n 1 n o + 0.5 n o 3 ( r 22 E 2 π 12 σ 2 ) n 2 n o 0.5 n o 3 ( r 22 E 2 + π 11 σ 2 ) n 3 n e 0.5 n e 3 π 31 σ 2 .
φ = 1 2 arctan [ 2 ( r 51 E 2 π 41 σ 2 ) ( 1 / n o 2 ) ( 1 / n e 2 ) + r 22 E 2 + ( π 11 π 31 ) σ 2 ] .
Δ n ( E 2 , σ 2 ) = n 2 n 1 n o 3 [ 0.5 ( π 12 π 11 ) σ 2 r 22 E 2 ] .
E 2 π 12 π 11 2 r 22 σ 2 ,
Δ n ( E 2 , σ 2 ) = n 2 n 3 n 2 2 sin 2 φ + n 3 2 cos 2 φ n 1 .
( 1 n o 2 + k 11 E 1 2 + π 11 σ 1 ) x 1 2 + ( 1 n o 2 + k 21 E 1 2 + π 21 σ 1 ) x 2 2 + ( 1 n o 2 + k 31 E 1 2 + π 31 σ 1 ) x 3 2 = 1 ,
{ π 11 = s 11 p 11 + 2 s 12 p 12 π 21 = π 31 = π 12 = s 11 p 12 + s 12 p 11 + s 12 p 12 .
{ n 1 n o 0.5 n o 3 ( k 11 E 1 2 + π 11 σ 1 ) n 2 = n 3 n o 0.5 n o 3 ( k 12 E 1 2 + π 12 σ 1 ) .
Δ n ( E 1 , σ 1 ) = ( n 2 n 1 ) or ( n 3 n 1 ) 0.5 n o 3 [ ( k 11 k 12 ) E 1 2 + ( π 11 π 12 ) σ 1 ] .
E 1 2 π 12 π 11 k 11 k 12 σ 1 = ( p 11 p 12 ) ( s 12 s 11 ) k 11 k 12 σ 1 ,
I o = 0.5 I i [ 1 ± cos ( Δ γ ) ] ,
Δ γ ( E 2 , σ 2 ) = 2 π λ Δ n ( E 2 , σ 2 ) ( 1 + s 23 σ 2 ) L ,
σ 2 = σ 20 + ε 2 s 11 σ 20 + k ε 3 p s 11 = σ 20 + k d 33 u 1 t s 11 ,
I o = 0.5 I i [ 1 ± sin ( Δ γ ) ] .
0 L E 2 ( x 3 ) d x 3 = π 12 π 11 2 r 22 0 L σ 2 ( x 3 ) d x 3 ,
0 L E 1 2 ( x 3 ) d x 3 = π 12 π 11 k 11 k 12 0 L σ 1 ( x 3 ) d x 3 ,
( 1 n o 2 + π 12 σ 2 ) x 1 2 + ( 1 n o 2 + π 22 σ 2 ) x 2 2 + ( 1 n e 2 + π 32 σ 2 ) x 3 2 + 2 π 42 σ 2 x 2 x 3 + 2 r 51 E 1 x 3 x 1 2 r 61 E 1 x 1 x 2 = 1 ,

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