Abstract

Optical current sensors based on polarization-rotated reflection interferometry are demonstrated using polymeric integrated optics and various functional optical waveguide devices. Interferometric sensors normally require bias feedback control for maintaining the operating point, which increases the cost. In order to resolve this constraint of feedback control, a multimode interference (MMI) waveguide device is integrated onto the current-sensor optical chip in this work. From the multiple outputs of the MMI, a 90° phase-shifted transfer function is obtained. Using passive quadrature demodulation, we demonstrate that the sensor could maintain the output signal regardless of the drift in the operating bias-point.

© 2016 Optical Society of America

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References

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  1. K. Bohnert, P. Gabus, J. Nehring, H. Brändle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25(11), 3602–3609 (2007).
    [Crossref]
  2. J. D. P. Hrabliuk, “Optical current sensors eliminate CT saturation,” in Proceedings of IEEE conference on PES Winter Meeting (IEEE, 2002) 2, pp.1478–2481.
  3. A. Enokihara, M. Izutsu, and T. Sueta, “Optical fiber sensors using the method of polarization-rotated reflection,” J. Lightwave Technol. 5(11), 1584–1590 (1987).
    [Crossref]
  4. K. Bohnert, P. Gabus, J. Nehring, and H. Brandle, “Temperature and vibration insensive fiber-optic current sensor,” J. Lightwave Technol. 20(2), 267–276 (2002).
    [Crossref]
  5. T. Masao, K. Sasaki, and K. Terai, “Optical current sensor for DC measurement,” in Proceedings of IEEE conference on Asia Pacific IEEE/PES Transmiss. Distrib. Conf. Exhibit. (IEEE, 2002) 1, pp.440–443.
  6. R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
    [Crossref]
  7. F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).
  8. J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
    [Crossref] [PubMed]
  9. K. Kurosawa, “Development of fiber-optic current sensing technique and its applications in electric power systerms,” Photon. Sensors 4(1), 12–20 (2014).
    [Crossref]
  10. G. M. Muler, L. Yang, A. Frank, and K. Bohnert, “Simple Fiber-optic current sensor with integrated-optics polarization splitter for interrogation,” in Applied Industrial Optics: Spectroscopy, Imaging and Metrology Conference, 2014 OSA Technical Digest Series (Optical Society of America, 2014), paper AM4A.3.
    [Crossref]
  11. D. W. Stowe and T.-Y. Hsu, “Demodulation of interferometric sensors using a fiber-optic passive quadrature demodulator,” J. Lightwave Technol. 1(3), 519–523 (1983).
    [Crossref]
  12. Y. Li and R. Baets, “Homodyne laser Doppler vibrometer on silicon-on-insulator with integrated 90 degree optical hybrids,” Opt. Express 21(11), 13342–13350 (2013).
    [Crossref] [PubMed]
  13. 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]
  14. W.-S. Chu, S.-M. Kim, and M.-C. Oh, “Integrated optic current transducers incorporating photonic crystal fiber for reduced temperature dependence,” Opt. Express 23(17), 22816–22825 (2015).
    [Crossref] [PubMed]
  15. M.-C. Oh, J.-K. Seo, K.-J. Kim, H. Kim, J.-W. Kim, and W.-S. Chu, “Optical current sensors consisting of polymeric waveguide components,” J. Lightwave Technol. 28(12), 1851–1857 (2010).
    [Crossref]
  16. L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
    [Crossref]
  17. Y. Sun, X. Jiang, and M. Wang, “Analysis of imaging properties in multimode interference couplers,” Proc. SPIE 5279, 192–198 (2004).
    [Crossref]
  18. L. B. Soldano and C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
    [Crossref]

2015 (1)

2014 (1)

K. Kurosawa, “Development of fiber-optic current sensing technique and its applications in electric power systerms,” Photon. Sensors 4(1), 12–20 (2014).
[Crossref]

2013 (2)

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Y. Li and R. Baets, “Homodyne laser Doppler vibrometer on silicon-on-insulator with integrated 90 degree optical hybrids,” Opt. Express 21(11), 13342–13350 (2013).
[Crossref] [PubMed]

2012 (1)

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

2007 (1)

2004 (1)

Y. Sun, X. Jiang, and M. Wang, “Analysis of imaging properties in multimode interference couplers,” Proc. SPIE 5279, 192–198 (2004).
[Crossref]

2002 (1)

2000 (1)

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

1995 (1)

L. B. Soldano and C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

1987 (1)

A. Enokihara, M. Izutsu, and T. Sueta, “Optical fiber sensors using the method of polarization-rotated reflection,” J. Lightwave Technol. 5(11), 1584–1590 (1987).
[Crossref]

1983 (1)

D. W. Stowe and T.-Y. Hsu, “Demodulation of interferometric sensors using a fiber-optic passive quadrature demodulator,” J. Lightwave Technol. 1(3), 519–523 (1983).
[Crossref]

Aldabaldetreku, G.

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Baets, R.

Baptista, J. M.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Bohnert, K.

Brandle, H.

Brändle, H.

Briffod, F.

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

Brunzel, M. G.

Casado, L.

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Chu, W.-S.

Durana, G.

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Enokihara, A.

A. Enokihara, M. Izutsu, and T. Sueta, “Optical fiber sensors using the method of polarization-rotated reflection,” J. Lightwave Technol. 5(11), 1584–1590 (1987).
[Crossref]

Frazao, O.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Gabus, P.

Hrabliuk, J. D. P.

J. D. P. Hrabliuk, “Optical current sensors eliminate CT saturation,” in Proceedings of IEEE conference on PES Winter Meeting (IEEE, 2002) 2, pp.1478–2481.

Hsu, T.-Y.

D. W. Stowe and T.-Y. Hsu, “Demodulation of interferometric sensors using a fiber-optic passive quadrature demodulator,” J. Lightwave Technol. 1(3), 519–523 (1983).
[Crossref]

Izutsu, M.

A. Enokihara, M. Izutsu, and T. Sueta, “Optical fiber sensors using the method of polarization-rotated reflection,” J. Lightwave Technol. 5(11), 1584–1590 (1987).
[Crossref]

Jiang, X.

Y. Sun, X. Jiang, and M. Wang, “Analysis of imaging properties in multimode interference couplers,” Proc. SPIE 5279, 192–198 (2004).
[Crossref]

Jorge, P.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Kim, H.

Kim, J.-W.

Kim, K.-J.

Kim, S.-M.

Kung, A.

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

Kurosawa, K.

K. Kurosawa, “Development of fiber-optic current sensing technique and its applications in electric power systerms,” Photon. Sensors 4(1), 12–20 (2014).
[Crossref]

Li, Y.

Martins, H.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Montero, A.

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Nascimento, I.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Nehring, J.

Nicati, P.-A.

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

Oh, M.-C.

Pennings, C. M.

L. B. Soldano and C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Petermann, K.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

Ribeiro, A. L.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Robert, P. A.

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

Santos, J. L.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Seo, J.-K.

Silva, R. M.

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

Soldano, L. B.

L. B. Soldano and C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Stowe, D. W.

D. W. Stowe and T.-Y. Hsu, “Demodulation of interferometric sensors using a fiber-optic passive quadrature demodulator,” J. Lightwave Technol. 1(3), 519–523 (1983).
[Crossref]

Sueta, T.

A. Enokihara, M. Izutsu, and T. Sueta, “Optical fiber sensors using the method of polarization-rotated reflection,” J. Lightwave Technol. 5(11), 1584–1590 (1987).
[Crossref]

Sun, Y.

Y. Sun, X. Jiang, and M. Wang, “Analysis of imaging properties in multimode interference couplers,” Proc. SPIE 5279, 192–198 (2004).
[Crossref]

Thevenaz, L.

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

Voigt, K.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

Wang, M.

Y. Sun, X. Jiang, and M. Wang, “Analysis of imaging properties in multimode interference couplers,” Proc. SPIE 5279, 192–198 (2004).
[Crossref]

Weinert, C. M.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

Winzer, G.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

Zimmermann, L.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

Zubia, E.

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Zubia, J.

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Appl. Sci. (1)

R. M. Silva, H. Martins, I. Nascimento, J. M. Baptista, A. L. Ribeiro, J. L. Santos, P. Jorge, and O. Frazao, “Optical current sensors for high power systems: a review,” Appl. Sci. 2(4), 602–628 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (1)

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-Band optical 90-hybrids based on silicon-on-insulator 4x4 waveguide couplers,” IEEE Photon. Technol. Lett. 21(3), 143–145 (2009).
[Crossref]

IEICE Trans. Electron. (1)

F. Briffod, L. Thevenaz, P.-A. Nicati, A. Kung, and P. A. Robert, “Polarimetric current sensor using an in-line faraday rotator,” IEICE Trans. Electron. E83-C(3), 331–335 (2000).

J. Lightwave Technol. (6)

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

A. Enokihara, M. Izutsu, and T. Sueta, “Optical fiber sensors using the method of polarization-rotated reflection,” J. Lightwave Technol. 5(11), 1584–1590 (1987).
[Crossref]

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

M.-C. Oh, J.-K. Seo, K.-J. Kim, H. Kim, J.-W. Kim, and W.-S. Chu, “Optical current sensors consisting of polymeric waveguide components,” J. Lightwave Technol. 28(12), 1851–1857 (2010).
[Crossref]

L. B. Soldano and C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

D. W. Stowe and T.-Y. Hsu, “Demodulation of interferometric sensors using a fiber-optic passive quadrature demodulator,” J. Lightwave Technol. 1(3), 519–523 (1983).
[Crossref]

Opt. Express (3)

Photon. Sensors (1)

K. Kurosawa, “Development of fiber-optic current sensing technique and its applications in electric power systerms,” Photon. Sensors 4(1), 12–20 (2014).
[Crossref]

Proc. SPIE (1)

Y. Sun, X. Jiang, and M. Wang, “Analysis of imaging properties in multimode interference couplers,” Proc. SPIE 5279, 192–198 (2004).
[Crossref]

Sensors (Basel) (1)

J. Zubia, L. Casado, G. Aldabaldetreku, A. Montero, E. Zubia, and G. Durana, “Design and development of a low-cost optical current sensor,” Sensors (Basel) 13(10), 13584–13595 (2013).
[Crossref] [PubMed]

Other (3)

G. M. Muler, L. Yang, A. Frank, and K. Bohnert, “Simple Fiber-optic current sensor with integrated-optics polarization splitter for interrogation,” in Applied Industrial Optics: Spectroscopy, Imaging and Metrology Conference, 2014 OSA Technical Digest Series (Optical Society of America, 2014), paper AM4A.3.
[Crossref]

T. Masao, K. Sasaki, and K. Terai, “Optical current sensor for DC measurement,” in Proceedings of IEEE conference on Asia Pacific IEEE/PES Transmiss. Distrib. Conf. Exhibit. (IEEE, 2002) 1, pp.440–443.

J. D. P. Hrabliuk, “Optical current sensors eliminate CT saturation,” in Proceedings of IEEE conference on PES Winter Meeting (IEEE, 2002) 2, pp.1478–2481.

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Figures (8)

Fig. 1
Fig. 1

Schematic configuration of the optical current sensors consisting of an integrated optics and a fiber sensing coil.

Fig. 2
Fig. 2

(a) Transfer function of a typical optical interferometer, in which the output signal amplitude is depending on the operating bias point, and (b) Two transfer functions with 90° phase difference which are incorporated for the signal processing so as to eliminate the dependence of operating bias.

Fig. 3
Fig. 3

Fabrication procedure of the polymeric integrated optics chip.

Fig. 4
Fig. 4

(a) Delayed Mach-Zehnder interferometer connected to the MMI device for the purpose of phase delay characterization, (b) Simulation results of the MMI output spectrum for a MMI length of 5150 μm, and (c) Spectral response of the fabricated MMI with a length of 5230 μm.

Fig. 5
Fig. 5

(a) The relative optical output power of MMI with various multimode region length, and (b) the relative phase difference between the output signals, (c) and (d) show the BPM simulation results.

Fig. 6
Fig. 6

For a change of ϕb over 2π, (a) the time average power signals 〈I1〉 and 〈I2〉, and (b) the amplitude signals A1 and A2 were calculated from the sensor output signal. (c) a Lissajous curve drawn by the 〈I1〉 and 〈I2〉 for calculating the phase error.

Fig. 7
Fig. 7

The sensor output signals as a function of ϕb obtained by the simple magnitude of orthogonal vectors (green), the phase error compensation (black), and the power change calibration (red).

Fig. 8
Fig. 8

Bias-free OCT sensor output measured without the feedback bias control, in which the output signal exhibited good linearity with a peak error of ± 0.5%, and an RMS error of 0.2%.

Tables (1)

Tables Icon

Table 1 Comparison of the phase difference (Δϕ) and the phase error (ϕe) of the output transfer curves obtained from simulation and measurement data.

Equations (21)

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

ϕ F ( t )= A F sinωt,
I= I 0 2 { 1+cos [ ϕ F ( t )+ ϕ b ] },
I= I 0 2 [ 1+ J 0 ( A F )cos ϕ b 2 J 1 ( A F )sin ϕ b sinωt+... ],
I= 1 T 0 T I dt= I 0 2 [ 1+ J 0 ( A F )cos φ b ] I 0 2 [ 1+cos φ b ],for A F <<1,
A O = I 0 J 1 ( A F )sin ϕ b I 0 A F sin ϕ b ,for A F <<1.
I 1 = I 0 2 [ 1+cos( ϕ b + A F sinωt ) ] = I 0 2 [ 1+ J 0 ( A F )cos ϕ b 2 J 1 ( A F )sin ϕ b sinωt+... ],
I 2 = I 0 2 [ 1+sin( ϕ b + A F sinωt ) ] = I 0 2 [ 1+ J 0 ( A F )sin ϕ b +2 J 1 ( A F )cos ϕ b sinωt+... ].
I 1 = I 0 2 [ 1+ J 0 ( A F )cos( ϕ b ) ],
I 2 = I 0 2 [ 1+ J 0 ( A F )sin( ϕ b ) ],
A 1 = I 0 J 1 ( A F )sin ϕ b ,
A 2 = I 0 J 1 ( A F )cos ϕ b .
A O = A 1 2 + A 2 2 = I 0 J 1 ( A F ) I 0 A F ,for A F <<1
I 1 = I 0 2 [ 1+cos( ϕ b + ϕ e + A F sinωt ) ] = I 0 2 [ 1+ J 0 ( A F )cos( ϕ b + ϕ e )2 J 1 ( ϕ 0 )sin( ϕ b + ϕ e )sinωt+ ],
I 2 = I 0 2 [ 1+sin( ϕ b + ϕ e + A F sinωt ) ] = I 0 2 [ 1+ J 0 ( A F )sin( ϕ b ϕ e )+2 J 1 ( A F )cos( ϕ b ϕ e )sinωt+.... ],
I 1 = I 0 2 [ 1+ J 0 ( A F )cos( ϕ b + ϕ e ) ],
I 2 = I 0 2 [ 1+ J 0 ( A F )sin( ϕ b ϕ e ) ],
A 1 = I 0 J 1 ( A F )sin( ϕ b + ϕ e ),
A 2 = I 0 J 1 ( A F )cos( ϕ b ϕ e ).
A O = A 1 ( I 2 I 0 2 )+ A 2 ( I 1 I 0 2 ) = I 0 2 2 J 0 ( A F ) J 1 ( A F )cos( φ e ) = I 0 2 2 A F cos( ϕ e ),for A F <<1.
Δϕ=[ k 0 ( λ 0 ) k 0 ( λ 0 +Δλ)] n eff Δl =( 1 λ 0 1 λ 0 +Δλ )2π n eff Δl,
Δλ= λ 0 2 n eff Δl λ 0

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