Abstract

A polarimetric current sensor based on polarization division multiplexing (PDM) detection is proposed. The novel sensor head with a heat insulation cavity only induces a small level of birefringence. Comparing with polarization diversity (PD) detection, the sensitivity of PDM detection is the double of PD detection. Moreover, PDM detection is more suitable in the presence of the phase modulation error. In addition, the noise and the shifting of the Verdet constant are proved to be the main influence factors of the sensor performance as the source power decline.

© 2014 Optical Society of America

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References

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  1. H. Y. Zhang, Y. K. Dong, J. Leeson, L. Chen, and X. Y. Bao, “High sensitivity optical fiber current sensor based on polarization diversity and a Faraday rotation mirror cavity,” Appl. Opt. 50(6), 924–929 (2011).
    [Crossref] [PubMed]
  2. B. C. Chu, Y. N. Ning, and D. A. Jackson, “Faraday current sensor that uses a triangular-shaped bulk-optic sensing element,” Opt. Lett. 17(16), 1167–1169 (1992).
    [Crossref] [PubMed]
  3. Y. N. Ning, B. C. Chu, and D. A. Jackson, “Miniature Faraday current sensor based on multiple critical angle reflections in a bulk-optic ring,” Opt. Lett. 16(24), 1996–1998 (1991).
    [Crossref] [PubMed]
  4. Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
    [Crossref]
  5. C. X. Zhang, C. S. Li, X. X. Wang, L. J. Li, J. Yu, and X. J. Feng, “Design principle for sensing coil of fiber-optic current sensor based on geometric rotation effect,” Appl. Opt. 51(18), 3977–3988 (2012).
    [Crossref] [PubMed]
  6. M. C. Oh, W. S. Chu, K. J. Kim, and J. W. Kim, “Polymer waveguide integrated-optic current transducers,” Opt. Express 19(10), 9392–9400 (2011).
    [Crossref] [PubMed]
  7. K. Bohnert, H. Brandle, M. G. Brunzel, P. Gabus, and P. Guggenbach, “Highly accurate fiber-optic DC current sensor for the electrowinning industry,” IEEE Trans. Ind. Appl. 43(1), 180–187 (2007).
    [Crossref]
  8. J. Blake, P. Tantaswadi, and R. T. De Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11(1), 116–121 (1996).
    [Crossref]
  9. K. Bohnert, P. Gabus, J. Nehring, and H. Brandle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
    [Crossref]
  10. H. Zhang, Y. S. Qiu, H. Li, A. X. Huang, H. X. Chen, and G. M. Li, “High-current-sensitivity all-fiber current sensor based on fiber loop architecture,” Opt. Express 20(17), 18591–18599 (2012).
    [Crossref] [PubMed]
  11. D. Alasia and L. Thevenaz, “A novel all-fibre configuration for a flexible polarimetric current sensor,” Meas. Sci. Technol. 15(8), 1525–1530 (2004).
    [Crossref]
  12. A. M. Smith, “Polarization and magnetooptic properties of single-mode optical fiber,” Appl. Opt. 17(1), 52–56 (1978).
    [Crossref] [PubMed]
  13. L. Bertolini, M. Carsana, and P. Pedeferri, “Corrosion behaviour of steel in concrete in the presence of stray current,” Corros. Sci. 49(3), 1056–1068 (2007).
    [Crossref]
  14. S. Y. Xu, W. Li, and Y. Q. Wang, “Effects of vehicle running mode on rail potential and stray current in DC mass transit systems,” IEEE T. Veh. Technol. 62(8), 3569–3580 (2013).
    [Crossref]
  15. P. Tantaswadi, “Simulation of birefringence effects in reciprocal fiber-optic polarimetric current sensor,” in Lightmetry: Metrology, Spectroscopy, and Testing Techniques Using Light (SPIE-Int. Soc. Optical Engineering, Bellingham, 2001), pp. 158–164.
  16. X. M. Liu, Magnetic Measurement (China Machine, 1989), Chap. 3.
  17. D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
    [Crossref]
  18. K. Bohnert, P. Gabus, J. Nehring, H. Brandle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
    [Crossref]
  19. Y. Namihira, “Opto-elastic constant in single mode optical fibers,” J. Lightwave Technol. 3(5), 1078–1083 (1985).
    [Crossref]
  20. C. Z. Tan and J. Arndt, “Faraday effect in silica glasses,” Phys. B 233(1), 1–7 (1997).
    [Crossref]
  21. Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
    [Crossref]

2013 (1)

S. Y. Xu, W. Li, and Y. Q. Wang, “Effects of vehicle running mode on rail potential and stray current in DC mass transit systems,” IEEE T. Veh. Technol. 62(8), 3569–3580 (2013).
[Crossref]

2012 (2)

2011 (2)

2007 (3)

K. Bohnert, H. Brandle, M. G. Brunzel, P. Gabus, and P. Guggenbach, “Highly accurate fiber-optic DC current sensor for the electrowinning industry,” IEEE Trans. Ind. Appl. 43(1), 180–187 (2007).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, H. Brandle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
[Crossref]

L. Bertolini, M. Carsana, and P. Pedeferri, “Corrosion behaviour of steel in concrete in the presence of stray current,” Corros. Sci. 49(3), 1056–1068 (2007).
[Crossref]

2005 (1)

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

2004 (2)

D. Alasia and L. Thevenaz, “A novel all-fibre configuration for a flexible polarimetric current sensor,” Meas. Sci. Technol. 15(8), 1525–1530 (2004).
[Crossref]

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

2002 (1)

1997 (1)

C. Z. Tan and J. Arndt, “Faraday effect in silica glasses,” Phys. B 233(1), 1–7 (1997).
[Crossref]

1996 (1)

J. Blake, P. Tantaswadi, and R. T. De Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11(1), 116–121 (1996).
[Crossref]

1992 (1)

1991 (2)

Y. N. Ning, B. C. Chu, and D. A. Jackson, “Miniature Faraday current sensor based on multiple critical angle reflections in a bulk-optic ring,” Opt. Lett. 16(24), 1996–1998 (1991).
[Crossref] [PubMed]

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
[Crossref]

1985 (1)

Y. Namihira, “Opto-elastic constant in single mode optical fibers,” J. Lightwave Technol. 3(5), 1078–1083 (1985).
[Crossref]

1978 (1)

Alasia, D.

D. Alasia and L. Thevenaz, “A novel all-fibre configuration for a flexible polarimetric current sensor,” Meas. Sci. Technol. 15(8), 1525–1530 (2004).
[Crossref]

Arndt, J.

C. Z. Tan and J. Arndt, “Faraday effect in silica glasses,” Phys. B 233(1), 1–7 (1997).
[Crossref]

Bao, X. Y.

Bertolini, L.

L. Bertolini, M. Carsana, and P. Pedeferri, “Corrosion behaviour of steel in concrete in the presence of stray current,” Corros. Sci. 49(3), 1056–1068 (2007).
[Crossref]

Blake, J.

J. Blake, P. Tantaswadi, and R. T. De Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11(1), 116–121 (1996).
[Crossref]

Bohnert, K.

Brandle, H.

Brunzel, M. G.

K. Bohnert, H. Brandle, M. G. Brunzel, P. Gabus, and P. Guggenbach, “Highly accurate fiber-optic DC current sensor for the electrowinning industry,” IEEE Trans. Ind. Appl. 43(1), 180–187 (2007).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, H. Brandle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
[Crossref]

Carsana, M.

L. Bertolini, M. Carsana, and P. Pedeferri, “Corrosion behaviour of steel in concrete in the presence of stray current,” Corros. Sci. 49(3), 1056–1068 (2007).
[Crossref]

Chen, H. X.

Chen, L.

Chu, B. C.

Chu, W. S.

Day, G. W.

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
[Crossref]

De Carvalho, R. T.

J. Blake, P. Tantaswadi, and R. T. De Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11(1), 116–121 (1996).
[Crossref]

Dong, Y. K.

Etzel, S. M.

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
[Crossref]

Feng, R. Y.

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Feng, X. J.

Gabus, P.

Guggenbach, P.

K. Bohnert, H. Brandle, M. G. Brunzel, P. Gabus, and P. Guggenbach, “Highly accurate fiber-optic DC current sensor for the electrowinning industry,” IEEE Trans. Ind. Appl. 43(1), 180–187 (2007).
[Crossref]

Huang, A. X.

Huang, Z. J.

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Jackson, D. A.

Jarvis, R. A.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

Kim, J. W.

Kim, K. J.

Leeson, J.

Li, C. S.

Li, G. M.

Li, H.

Li, L. J.

Li, Q. B.

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Li, W.

S. Y. Xu, W. Li, and Y. Q. Wang, “Effects of vehicle running mode on rail potential and stray current in DC mass transit systems,” IEEE T. Veh. Technol. 62(8), 3569–3580 (2013).
[Crossref]

Luther-Davies, B.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

Madden, S.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

Namihira, Y.

Y. Namihira, “Opto-elastic constant in single mode optical fibers,” J. Lightwave Technol. 3(5), 1078–1083 (1985).
[Crossref]

Nehring, J.

Ning, Y. N.

Oh, M. C.

Pedeferri, P.

L. Bertolini, M. Carsana, and P. Pedeferri, “Corrosion behaviour of steel in concrete in the presence of stray current,” Corros. Sci. 49(3), 1056–1068 (2007).
[Crossref]

Qiu, Y. S.

Rode, A. V.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

Rose, A. H.

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
[Crossref]

Ruan, Y.

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

Shi, J. H.

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Smith, A. M.

Tan, C. Z.

C. Z. Tan and J. Arndt, “Faraday effect in silica glasses,” Phys. B 233(1), 1–7 (1997).
[Crossref]

Tang, D.

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
[Crossref]

Tantaswadi, P.

J. Blake, P. Tantaswadi, and R. T. De Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11(1), 116–121 (1996).
[Crossref]

Thevenaz, L.

D. Alasia and L. Thevenaz, “A novel all-fibre configuration for a flexible polarimetric current sensor,” Meas. Sci. Technol. 15(8), 1525–1530 (2004).
[Crossref]

Wang, H. L.

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Wang, X. X.

Wang, Y. Q.

S. Y. Xu, W. Li, and Y. Q. Wang, “Effects of vehicle running mode on rail potential and stray current in DC mass transit systems,” IEEE T. Veh. Technol. 62(8), 3569–3580 (2013).
[Crossref]

Wang, Z. P.

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Xu, S. Y.

S. Y. Xu, W. Li, and Y. Q. Wang, “Effects of vehicle running mode on rail potential and stray current in DC mass transit systems,” IEEE T. Veh. Technol. 62(8), 3569–3580 (2013).
[Crossref]

Yu, J.

Zhang, C. X.

Zhang, H.

Zhang, H. Y.

Appl. Opt. (3)

Corros. Sci. (1)

L. Bertolini, M. Carsana, and P. Pedeferri, “Corrosion behaviour of steel in concrete in the presence of stray current,” Corros. Sci. 49(3), 1056–1068 (2007).
[Crossref]

IEEE T. Veh. Technol. (1)

S. Y. Xu, W. Li, and Y. Q. Wang, “Effects of vehicle running mode on rail potential and stray current in DC mass transit systems,” IEEE T. Veh. Technol. 62(8), 3569–3580 (2013).
[Crossref]

IEEE Trans. Ind. Appl. (1)

K. Bohnert, H. Brandle, M. G. Brunzel, P. Gabus, and P. Guggenbach, “Highly accurate fiber-optic DC current sensor for the electrowinning industry,” IEEE Trans. Ind. Appl. 43(1), 180–187 (2007).
[Crossref]

IEEE Trans. Power Deliv. (1)

J. Blake, P. Tantaswadi, and R. T. De Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11(1), 116–121 (1996).
[Crossref]

J. Lightwave Technol. (4)

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fiber coils: application to optical fiber current sensors,” J. Lightwave Technol. 9(8), 1031–1037 (1991).
[Crossref]

Y. Namihira, “Opto-elastic constant in single mode optical fibers,” J. Lightwave Technol. 3(5), 1078–1083 (1985).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, and H. Brandle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
[Crossref]

K. Bohnert, P. Gabus, J. Nehring, H. Brandle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
[Crossref]

Meas. Sci. Technol. (1)

D. Alasia and L. Thevenaz, “A novel all-fibre configuration for a flexible polarimetric current sensor,” Meas. Sci. Technol. 15(8), 1525–1530 (2004).
[Crossref]

Opt. Commun. (1)

Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

Z. P. Wang, Q. B. Li, R. Y. Feng, H. L. Wang, Z. J. Huang, and J. H. Shi, “Effects of the polarizer parameters upon the performance of an optical current sensor,” Opt. Laser Technol. 36(2), 145–149 (2004).
[Crossref]

Opt. Lett. (2)

Phys. B (1)

C. Z. Tan and J. Arndt, “Faraday effect in silica glasses,” Phys. B 233(1), 1–7 (1997).
[Crossref]

Other (2)

P. Tantaswadi, “Simulation of birefringence effects in reciprocal fiber-optic polarimetric current sensor,” in Lightmetry: Metrology, Spectroscopy, and Testing Techniques Using Light (SPIE-Int. Soc. Optical Engineering, Bellingham, 2001), pp. 158–164.

X. M. Liu, Magnetic Measurement (China Machine, 1989), Chap. 3.

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Figures (7)

Fig. 1
Fig. 1 Optical configuration of this sensor.
Fig. 2
Fig. 2 Configuration of the sensor head: 1. Sensing fiber; 2.Solenoid; 3.Framework; 4. Metal casings; 5. Heat insulation cavity; 6. L-form holder.
Fig. 3
Fig. 3 Simulation result of the sensor sensitivity.
Fig. 4
Fig. 4 Experimental result of the sensor sensitivity.
Fig. 5
Fig. 5 Experimental output in presence of the modulation error: (a) 15 Hz; (b) 30 Hz.
Fig. 6
Fig. 6 The original signals.
Fig. 7
Fig. 7 The de-noised signal.

Tables (2)

Tables Icon

Table 1 Result with PDM and PD detection

Tables Icon

Table 2 Similarity Coefficient

Equations (17)

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

L p =[ 1 0 0 0 ]. L m =[ 1 0 0 1 ]. L R =[ cos π 4 sin π 4 sin π 4 cos π 4 ]. L f1 =[ A+iB C C AiB ]. L f2 =[ A +i B C C A i B ]. A=cosp. A =cosq. B= δ 2 ( sinp p ). B = δ 2 ( sinq q ). C=( T+F )( sinp p ). C =( TF )( sinq q ). p= ( T+F ) 2 + ( δ/2 ) 2 .q= ( TF ) 2 + ( δ/2 ) 2 .
E o = L R L f2 L m L f1 L p E i =[ ( A A ' B B ' +C C ' +A C ' A ' C )i( A ' B+A B ' +B C ' + B ' C ) ( A A ' B B ' +C C ' A C ' + A ' C )+i( A ' B+A B ' B C ' B ' C ) ] 2 E xi 2 .
{ J x1 = E xi 2 2 [ ( A A ' B B ' +C C ' +A C ' A ' C ) 2 + ( A ' B+A B ' +B C ' + B ' C ) 2 ] J y1 = E xi 2 2 [ ( A A ' B B ' +C C ' A C ' + A ' C ) 2 + ( A ' B+A B ' B C ' B ' C ) 2 ] .
{ J x0 = E xi 2 2 [ ( A 0 2 B 0 2 + C 0 2 ) 2 + ( 2 A 0 B 0 +2 B 0 C 0 ) 2 ] J y0 = E xi 2 2 [ ( A 0 2 B 0 2 + C 0 2 ) 2 + ( 2 A 0 B 0 2 B 0 C 0 ) 2 ] .
A 0 =cosk. B 0 = δ 2 ( sink k ). C 0 =T( sink k ).k= ( T ) 2 + ( δ/2 ) 2 .
D 1 = J y1 J x1 J y1 + J x1 ; D 2 = J y1 J y0 J y0 J x1 J x0 J x0 .
H( t )= n 1 n 2 ( L/2 )ln b+ b 2 + ( L/2 ) 2 a+ a 2 + ( L/2 ) 2 i( t ).
F( t 0 )=VNlH( t 0 )=( VNl n 1 n 2 ( L/2 )ln b+ b 2 + ( L/2 ) 2 a+ a 2 + ( L/2 ) 2 )i( t 0 )=mi( t 0 ).
δ b = π λ EC r 2 R 2 .
L R =[ cos( π 4 +θ ) sin( π 4 +θ ) sin( π 4 +θ ) cos( π 4 +θ ) ].
{ J x0 = E xi 2 2 [ 1sin( 2θ ) ] J y0 = E xi 2 2 [ 1+sin( 2θ ) ] J x1 = E xi 2 2 [ 1sin( 4F+2θ ) ] J y1 = E xi 2 2 [ 1+sin( 4F+2θ ) ] .
D 1 = J y1 J x1 J y1 + J x1 =sin( 4F+2θ )4F+2θ.
D 2 = J y1 J y0 J y0 J x1 J x0 J x0 = sin( 4F+2θ )sin( 2θ ) 1+sin( 2θ ) sin( 4F+2θ )+sin( 2θ ) 1sin( 2θ ) 4F 1+2θ + 4F 12θ = 8F 14 θ 2 .
r 2 = ( m i=1 m f i u i i=1 m f i i=1 m u i ) 2 ( m i=1 m f i 2 ( i=1 m f i ) 2 )( m i=1 m u i 2 ( i=1 m u i ) 2 ) .
V( w )= C 0 w 2 n ( w 0 2 w 2 ) 2 .
n 2 =1+ C 1 1 w 0 2 w 2 .
V( λ )= 4 π 2 v 2 C 0 C 1 2 ( n 2 1 ) 2 n 1 λ 2 .

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