Abstract

In this article, we propose an optical heterodyne common-path gyroscope which has common-path configuration and full-dynamic range. Different from traditional non-common-path optical heterodyne technique such as Mach-Zehnder or Michelson interferometers, we use a two-frequency laser light source (TFLS) which can generate two orthogonally polarized light with a beat frequency has a common-path configuration. By use of phase measurement, this optical heterodyne gyroscope not only has the capability to overcome the drawback of the traditional interferometric fiber optic gyro: lack for full-dynamic range, but also eliminate the total polarization rotation caused by SMFs. Moreover, we also demonstrate the potential of miniaturizing this gyroscope as a chip device. Theoretically, if we assume that the wavelength of the laser light is 1550nm, the SMFs are 250m in length, and the radius of the fiber ring is 3.5cm, the bias stability is 0.872 deg/hr.

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2009

C. J. Yu, C. E. Lin, H. K. Teng, C. C. Tsai, and C. Chou, “Dual-frequency paired polarization phase shifting ellipsometer,” Opt. Commun.282(8), 1516–1520 (2009).
[CrossRef]

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

2008

2007

2006

2005

J. Zheng, “All-fiber single-mode fiber frequency-modulated continuous-wave Sagnac gyroscope,” Opt. Lett.30(1), 17–19 (2005).
[CrossRef] [PubMed]

J. Zheng, “Differential birefringent fiber frequency-modulated continuous-wave Sagnac gyroscope,” IEEE Photon. Technol. Lett.17(7), 1498–1500 (2005).
[CrossRef]

B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(5), 056621 (2005).
[CrossRef] [PubMed]

2004

J. Zheng, “Birefringent fiber frequency-modulated continuous-wave Sagnac gyroscope,” Electron. Lett.40(24), 1520–1521 (2004).
[CrossRef]

J. Zheng, “Analysis of optical frequency-modulated continuous-wave interference,” Appl. Opt.43(21), 4189–4198 (2004).
[CrossRef] [PubMed]

1999

1998

Z. H. Xie, Z. A. Jiang, S. S. Jian, and W. B. Tao, “Theoretical study on resonance characteristics of fiber optic ring resonator in fiber-optical ring resonator gyroscope,” Proc. SPIE3552, 267–271 (1998).
[CrossRef]

M. Li, Q. Tian, E. Y. Zhang, M. Zhang, and Y. B. Laio, “A novel passive-ring-resonator-gyro (R-FOG) with a two-coupler ring,” Proc. SPIE3555, 358–362 (1998).
[CrossRef]

1997

H. C. Lefèvre, “Fundamentals of the interferometric fiber-optic gyroscope,” Opt. Rev.4(1), A20–A27 (1997).
[CrossRef]

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol.15(3), 466–473 (1997).
[CrossRef]

1996

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng.18(2–3), 161–163 (1996).
[CrossRef]

1991

1988

1986

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol.4(8), 1071–1089 (1986).
[CrossRef]

1985

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

1984

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fiber gyroscope,” Electron. Lett.20(9), 368–370 (1984).
[CrossRef]

1983

1982

1981

1977

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett.30(9), 478–480 (1977).
[CrossRef]

1976

1967

E. J. Post, “Sagnac effect,” Mod. Phys.39(2), 475–493 (1967).
[CrossRef]

J. Killpatrick, “The laser gyro,” IEEE Spectr.4(10), 44–55 (1967).
[CrossRef]

1913

G. Sagnac, “L'éther lumineux démontré par l'effet du vent du vent relatif d'éther dans un interféromètre en rotation uniforme,” C. R. Acad. Sci.95, 708–710 (1913).

Armelles, G.

Balsamo, S. R.

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett.30(9), 478–480 (1977).
[CrossRef]

Boag, A.

Booher, D.

Calle, A.

Chan, Y. J.

Chen, B.

Chen, C. C.

Chen, C. D.

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng.18(2–3), 161–163 (1996).
[CrossRef]

Chen, W.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Chien, H. T.

Chiu, M. H.

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng.18(2–3), 161–163 (1996).
[CrossRef]

Chiu, W. Y.

Choi, S. S.

Chou, C.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Cui, F.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Dai, F.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Ding, C.

Ezekiel, S.

F. Zarinetchi, S. P. Smith, and S. Ezekiel, “Stimulated Brillouin fiber-optic laser gyroscope,” Opt. Lett.16(4), 229–231 (1991).
[CrossRef] [PubMed]

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett.30(9), 478–480 (1977).
[CrossRef]

Findaldy, T.

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Han, K. G.

Harumoto, M.

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol.15(3), 466–473 (1997).
[CrossRef]

Higashiguchi, M.

Hotate, K.

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol.15(3), 466–473 (1997).
[CrossRef]

K. Hotate, N. Okuma, M. Higashiguchi, and N. Niwa, “Rotation detection by optical heterodyne fiber gyro with frequency output,” Opt. Lett.7(7), 331–333 (1982).
[CrossRef] [PubMed]

Hou, C. H.

Huang, H.-S.

Huang, T. W.

Jackson, D. A.

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fiber gyroscope,” Electron. Lett.20(9), 368–370 (1984).
[CrossRef]

Jian, S. S.

Z. H. Xie, Z. A. Jiang, S. S. Jian, and W. B. Tao, “Theoretical study on resonance characteristics of fiber optic ring resonator in fiber-optical ring resonator gyroscope,” Proc. SPIE3552, 267–271 (1998).
[CrossRef]

Jiang, Z. A.

Z. H. Xie, Z. A. Jiang, S. S. Jian, and W. B. Tao, “Theoretical study on resonance characteristics of fiber optic ring resonator in fiber-optical ring resonator gyroscope,” Proc. SPIE3552, 267–271 (1998).
[CrossRef]

Jin, Z. H.

Jo, J. C.

Kai, K.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Kersey, A. D.

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fiber gyroscope,” Electron. Lett.20(9), 368–370 (1984).
[CrossRef]

Killpatrick, J.

J. Killpatrick, “The laser gyro,” IEEE Spectr.4(10), 44–55 (1967).
[CrossRef]

Kim, D. H.

Kim, S.

Kintner, E. C.

Koch, C.

Koseki, H.

Laio, Y. B.

M. Li, Q. Tian, E. Y. Zhang, M. Zhang, and Y. B. Laio, “A novel passive-ring-resonator-gyro (R-FOG) with a two-coupler ring,” Proc. SPIE3555, 358–362 (1998).
[CrossRef]

Lechuga, L. M.

Lefèvre, H. C.

H. C. Lefèvre, “Fundamentals of the interferometric fiber-optic gyroscope,” Opt. Rev.4(1), A20–A27 (1997).
[CrossRef]

Lewin, A. C.

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fiber gyroscope,” Electron. Lett.20(9), 368–370 (1984).
[CrossRef]

Li, K.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Li, M.

M. Li, Q. Tian, E. Y. Zhang, M. Zhang, and Y. B. Laio, “A novel passive-ring-resonator-gyro (R-FOG) with a two-coupler ring,” Proc. SPIE3555, 358–362 (1998).
[CrossRef]

Lin, C. E.

C. J. Yu, C. E. Lin, H. K. Teng, C. C. Tsai, and C. Chou, “Dual-frequency paired polarization phase shifting ellipsometer,” Opt. Commun.282(8), 1516–1520 (2009).
[CrossRef]

Lin, C.-E.

Ma, G.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Ma, H. I.

Nee, S.-M. F.

Niwa, N.

Noda, J.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol.4(8), 1071–1089 (1986).
[CrossRef]

Ohtsuka, Y.

Okamoto, K.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol.4(8), 1071–1089 (1986).
[CrossRef]

Okuma, N.

Pavlath, G. A.

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Post, E. J.

E. J. Post, “Sagnac effect,” Mod. Phys.39(2), 475–493 (1967).
[CrossRef]

Sagnac, G.

G. Sagnac, “L'éther lumineux démontré par l'effet du vent du vent relatif d'éther dans un interféromètre en rotation uniforme,” C. R. Acad. Sci.95, 708–710 (1913).

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Sasaki, Y.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol.4(8), 1071–1089 (1986).
[CrossRef]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Sepúlveda, B.

Shaw, H. J.

Shorthill, R. W.

Smith, S. P.

Steinberg, B. Z.

B. Z. Steinberg and A. Boag, “Aplitting of microcavity degenerate modes in rotating photonic crystals - the miniature optical gyroscopes,” J. Opt. Soc. Am. B24(1), 142–151 (2007).
[CrossRef]

B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(5), 056621 (2005).
[CrossRef] [PubMed]

Su, D. C.

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng.18(2–3), 161–163 (1996).
[CrossRef]

Tao, W. B.

Z. H. Xie, Z. A. Jiang, S. S. Jian, and W. B. Tao, “Theoretical study on resonance characteristics of fiber optic ring resonator in fiber-optical ring resonator gyroscope,” Proc. SPIE3552, 267–271 (1998).
[CrossRef]

Teng, H. K.

C. J. Yu, C. E. Lin, H. K. Teng, C. C. Tsai, and C. Chou, “Dual-frequency paired polarization phase shifting ellipsometer,” Opt. Commun.282(8), 1516–1520 (2009).
[CrossRef]

Tian, Q.

M. Li, Q. Tian, E. Y. Zhang, M. Zhang, and Y. B. Laio, “A novel passive-ring-resonator-gyro (R-FOG) with a two-coupler ring,” Proc. SPIE3555, 358–362 (1998).
[CrossRef]

Tsai, C. C.

C. J. Yu, C. E. Lin, H. K. Teng, C. C. Tsai, and C. Chou, “Dual-frequency paired polarization phase shifting ellipsometer,” Opt. Commun.282(8), 1516–1520 (2009).
[CrossRef]

Tsarev, A. V.

Vali, V.

Wu, J.-S.

Wu, X.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Wu, Y. H.

Xiao, Q.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Xie, Z. H.

Z. H. Xie, Z. A. Jiang, S. S. Jian, and W. B. Tao, “Theoretical study on resonance characteristics of fiber optic ring resonator in fiber-optical ring resonator gyroscope,” Proc. SPIE3552, 267–271 (1998).
[CrossRef]

Yu, C. J.

C. J. Yu, C. E. Lin, H. K. Teng, C. C. Tsai, and C. Chou, “Dual-frequency paired polarization phase shifting ellipsometer,” Opt. Commun.282(8), 1516–1520 (2009).
[CrossRef]

Yu, C.-J.

Zarinetchi, F.

Zhang, E. Y.

M. Li, Q. Tian, E. Y. Zhang, M. Zhang, and Y. B. Laio, “A novel passive-ring-resonator-gyro (R-FOG) with a two-coupler ring,” Proc. SPIE3555, 358–362 (1998).
[CrossRef]

Zhang, M.

M. Li, Q. Tian, E. Y. Zhang, M. Zhang, and Y. B. Laio, “A novel passive-ring-resonator-gyro (R-FOG) with a two-coupler ring,” Proc. SPIE3555, 358–362 (1998).
[CrossRef]

Zhang, W.

K. Kai, W. Zhang, W. Chen, K. Li, F. Dai, F. Cui, X. Wu, G. Ma, and Q. Xiao, “The development of micro-gyroscope technology,” J. Micromech. Microeng.19(11), 113001 (2009).
[CrossRef]

Zhang, X. L.

Zheng, J.

J. Zheng, “Differential birefringent fiber frequency-modulated continuous-wave Sagnac gyroscope,” IEEE Photon. Technol. Lett.17(7), 1498–1500 (2005).
[CrossRef]

J. Zheng, “All-fiber single-mode fiber frequency-modulated continuous-wave Sagnac gyroscope,” Opt. Lett.30(1), 17–19 (2005).
[CrossRef] [PubMed]

J. Zheng, “Analysis of optical frequency-modulated continuous-wave interference,” Appl. Opt.43(21), 4189–4198 (2004).
[CrossRef] [PubMed]

J. Zheng, “Birefringent fiber frequency-modulated continuous-wave Sagnac gyroscope,” Electron. Lett.40(24), 1520–1521 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett.30(9), 478–480 (1977).
[CrossRef]

C. R. Acad. Sci.

G. Sagnac, “L'éther lumineux démontré par l'effet du vent du vent relatif d'éther dans un interféromètre en rotation uniforme,” C. R. Acad. Sci.95, 708–710 (1913).

Electron. Lett.

J. Zheng, “Birefringent fiber frequency-modulated continuous-wave Sagnac gyroscope,” Electron. Lett.40(24), 1520–1521 (2004).
[CrossRef]

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fiber gyroscope,” Electron. Lett.20(9), 368–370 (1984).
[CrossRef]

IEEE Photon. Technol. Lett.

J. Zheng, “Differential birefringent fiber frequency-modulated continuous-wave Sagnac gyroscope,” IEEE Photon. Technol. Lett.17(7), 1498–1500 (2005).
[CrossRef]

IEEE Spectr.

J. Killpatrick, “The laser gyro,” IEEE Spectr.4(10), 44–55 (1967).
[CrossRef]

J. Lightwave Technol.

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol.15(3), 466–473 (1997).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Optical setup of TFLS.

Fig. 2
Fig. 2

The experimental result of TFLS. The sawtooth wave is generated by a function generator and amplified by a high voltage amplifier. The modulated signal is a Sinusoid wave.

Fig. 3
Fig. 3

The optical setup of the common-path heterodyne gyroscope is illustrated. Where PBS is polarizing beam splitter; BS is beam splitter; EOM is electro-optic modulator; BPF is band-pass filter; HWP is half-wave plate; PR is polarization rotator; DM is amplitude demodulator; PC is personal computer; P is polarizer; D is photodetector.

Fig. 4
Fig. 4

The arrangement of the integrated-optic devices on a chip (dash line) for heterodyne fiber-optical gyroscope.

Tables (1)

Tables Icon

Table 1 According to the plus and minus symbols of Eqs. (24) and (29), we can determine the quadrant of the measured phase shift

Equations (35)

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E L =( 1 0 ) A 0 expi( ω 0 t k 0 z).
E in =P( 45° )EO( ωt )P( 45° ) E L = A 0 ( 1 1 )cos( ωt/2 )expi( ω 0 t k 0 z ).
G=R S 12 R+T S 21 T = G s + G p .
R= expi( π/4 ) 2 ( 0 0 0 1 ),
T= expi( π/4 ) 2 ( 1 0 0 0 ),
S 12 =exp( i[ ϕ _ ( L )+ ϕ ccw ] )( cosθ( L )exp( i ξ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) cosθ( L )exp( i ξ( L ) 2 ) ),
S 21 =exp( i[ ϕ _ ( L )+ ϕ cw ] )( cosθ( L )exp( i ξ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) cosθ( L )exp( i ξ( L ) 2 ) ),
E out p = G p E in =T S 21 T E in = expi( π/2 ) 2 exp[ i( ϕ ¯ ( L )+ ϕ cw ) ]( 1 0 0 0 ) ×( cosθ( L )exp( i ξ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) cosθ( L )exp( i ξ( L ) 2 ) )( 1 0 0 0 ) E in = expi( π/2 ) 2 exp[ i( ϕ ¯ ( L )+ ϕ cw ) ]( cosθ( L )exp( i ξ( L ) 2 ) 0 0 0 ) E in = 1 2 A 0 exp[ i( ϕ cw + ξ( L ) 2 + π 2 ) ]( 1 0 )cosθ( L )cos( ωt /2 )exp{ i[ ω 0 t k 0 z+ ϕ ¯ ( L ) ] },
E out s = G s E in =R S 12 R E in = expi( π /2 ) 2 exp[ i( ϕ ¯ ( L )+ ϕ ccw ) ]( 0 0 0 1 ) ×( cosθ( L )exp( i ξ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) sinθ( L )exp( i ϕ( L ) 2 ) cosθ( L )exp( i ξ( L ) 2 ) )( 0 0 0 1 ) E in = expi( π/2 ) 2 exp[ i( ϕ ¯ ( L )+ ϕ ccw ) ]( 0 0 0 cosθ( L )exp( i ξ( L ) 2 ) ) E in = 1 2 A 0 exp[ i( ϕ ccw ξ( L ) 2 π 2 ) ]( 0 1 )cosθ( L )cos( ωt /2 )expi( ω 0 t k 0 z+ ϕ ¯ ( L ) ).
E out p1 = 2 4 A 0 exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 1 ) ]( 1 0 )cosθ( L )cos( ωt /2 )expi[ ω 0 t k 0 z+ ϕ ¯ ( L ) ],
E out s1 = 2 4 A 0 exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 1 ) ]( 0 1 )cosθ( L )cos( ωt /2 )expi[ ω 0 t k 0 z+ ϕ ¯ ( L ) ].
E out p2 = 2 4 A 0 exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 2 ) ]( 1 0 )cosθ( L )cos( ωt /2 )expi[ ω 0 t k 0 z+ ϕ ¯ ( L ) ],
E out s2 = 2 4 A 0 exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 2 ) ]( 0 1 )cosθ( L )cos( ωt /2 )expi[ ω 0 t k 0 z+ ϕ ¯ ( L ) ].
E out p1 + E out s1 = 1 4 A 0 cosθ( L )cos( ωt /2 )expi[ ω 0 t k 0 z+ ϕ ¯ ( L ) ] ×{ exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 1 ) ]+exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 1 ) ] }.
E out p2 + E out s2 = 1 4 A 0 cosθ( L )cos( ωt /2 )expi[ ω 0 t+ k 0 z+ ϕ ¯ ( L ) ] ×{ exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 2 ) ]+exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 2 ) ] }.
I 1 = | E out p1 + E out s1 | 2 = 1 16 A 0 2 cos 2 θ( L ) cos 2 ( ωt /2 ) { exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 1 ) ] +exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 1 ) ] } 2 = 1 16 A 0 2 cos 2 θ( L ) cos 2 ( ωt /2 ) ×{ exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 1 ) ] +exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 1 ) ] }{ exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 1 ) ] +exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 1 ) ] } = 1 16 A 0 2 cos 2 θ( L ) cos 2 ( ωt /2 )[ 2+2cos( ϕ s +ξ( L )+π+2 δ 1 ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ωt ) ][ 1+cos( ϕ s +ξ( L )+π+2 δ 1 ) ].
I 1 AC = 1 16 A 0 2 cos 2 θ( L )cos( ωt )[ 1+cos( ϕ s +ξ( L )+π+2 δ 1 ) ],
I 2 = | E out p2 + E out s2 | 2 = 1 16 A 0 2 cos 2 θ( L ) cos 2 ( ωt /2 ) { exp[ i( ϕ cw + ξ( L ) 2 + π 2 + δ 1 ) ] +exp[ i( ϕ ccw ξ( L ) 2 π 2 δ 1 ) ] } 2 = 1 16 A 0 2 cos 2 θ( L ) cos 2 ( ωt /2 )[ 2+2cos( ϕ s +ξ( L )+π+2 δ 2 ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ωt ) ][ 1+cos( ϕ s +ξ( L )+π+2 δ 2 ) ].
I 2 AC = 1 16 A 0 2 cos 2 θ( L )cos( ωt )[ 1+cos( ϕ s +ξ( L )+π+2 δ 1 ) ].
I 1 cos,AC = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L )+π ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1cos( ϕ s +ξ( L ) ) ],
I 2 cos,AC = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L )+ππ ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L ) ) ],
I 2 cos,AC I 1 cos,AC = 1 16 A 0 2 cos 2 θ( L ){ [ 1+cos( ϕ s +ξ( L ) ) ][ 1cos( ϕ s +ξ( L ) ) ] } = 1 8 A 0 2 cos 2 θ( L )cos( ϕ s +ξ( L ) ),
I 2 cos,AC + I 1 cos,AC = 1 16 A 0 2 cos 2 θ( L ){ [ 1+cos( ϕ s +ξ( L ) ) ]+[ 1cos( ϕ s +ξ( L ) ) ] } = 1 8 A 0 2 cos 2 θ( L ),
C= I 2 cos,AC I 1 cos,AC I 2 cos,AC + I 1 cos,AC =cos( ϕ s +ξ( L ) ).
I 1 sin,AC = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L )+π π 2 ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L )+ π 2 ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1sin( ϕ s +ξ( L ) ) ],
I 2 sin,AC = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L )+π+ π 2 ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1+cos( ϕ s +ξ( L )+ 3π 2 ) ] = 1 16 A 0 2 cos 2 θ( L )[ 1+sin( ϕ s +ξ( L ) ) ].
I 2 sin,AC I 1 sin,AC = 1 16 A 0 2 cos 2 θ( L ){ [ 1+sin( ϕ s +ξ( L ) ) ][ 1sin( ϕ s +ξ( L ) ) ] } = 1 8 A 0 2 cos 2 θ( L )sin( ϕ s +ξ( L ) ),
I 2 sin,AC + I 1 sin,AC = 1 16 A 0 2 cos 2 θ( L ){ [ 1+sin( ϕ s +ξ( L ) ) ]+[ 1sin( ϕ s +ξ( L ) ) ] } = 1 8 A 0 2 cos 2 θ( L ),
S= I 2 sin,AC I 1 sin,AC I 2 sin,AC + I 1 sin,AC =sin( ϕ s +ξ( L ) ).
ϕ s = 8 π 2 R 2 Δω λc = tan 1 ( S C )ξ( L ) tan 1 ( I 2 sin,AC I 1 sin,AC I 2 cos,AC I 1 cos,AC )ξ( L ),
ϕ s tan 1 ( I 2 sin,AC I 1 sin,AC I 2 cos,AC I 1 cos,AC )ξ( L ).
ϕ s signal +Δ ϕ s drift = tan 1 ( ( I 2 sin,AC +Δ I 2 sin,AC )( I 1 sin,AC +Δ I 1 sin,AC ) ( I 2 cos,AC +Δ I 2 cos,AC )( I 1 cos,AC +Δ I 1 cos,AC ) )ξ( L ) = tan 1 ( ( I 2 sin,AC I 1 sin,AC )+( Δ I 2 sin,AC Δ I 1 sin,AC ) ( I 2 cos,AC I 1 cos,AC )+( Δ I 2 cos,AC Δ I 1 cos,AC ) )ξ( L ).
Δ ϕ s drift = tan 1 ( ( Δ I 2 sin,AC Δ I 1 sin,AC ) ( I 2 cos,AC I 1 cos,AC )+( Δ I 2 cos,AC Δ I 1 cos,AC ) )ξ( L ).
Δ ϕ s drift  =ta n 1 [ ( Δ I 2 sin,AC Δ I 1 sin,AC ) ( I 2 cos,AC I 1 cos,AC ) ]ξ( L ) = 8 π 2 R 2 δω λc ξ( L ),
δω= λc 8 π 2 R 2 tan 1 [ ( Δ I 2 sin,AC Δ I 1 sin,AC ) ( I 2 cos,AC I 1 cos,AC ) ].

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