Abstract

In order to reduce the nonlinearity caused by an error of phase modulation depth, carrier phase delay and non-ideal performance of the low pass filters in the sinusoidal phase modulating interferometer (SPMI), a modified EOM-based SPMI is proposed in this paper to realize real-time normalization of the quadrature components for the arctangent approach of phase generated carrier (PGC-Arctan) demodulation. To verify the effectiveness of the real-time normalization technique, a fixed-phase-difference detection method is presented to evaluate the periodic nonlinearity in real time. The modified EOM-based SPMI is consisted of a monitor interferometer and a probe interferometer. The two interferometers share a reference corner cube, which is mounted on a slowly moving stage, thus periodic interference signals are generated for real-time normalization of the quadrature components in PGC demodulation. Subtracting the demodulated phase of the monitor interferometer from the phase of the probe interferometer, the phase to be measured can be obtained. The fixed-phase-difference detection method is realized by detecting an interference signal with two photodetectors, which are placed at an interval of quarter fringe, and the variation of the fixed-phase-difference can reflect the nonlinear error in PGC demodulation. Experiments of real-time normalization, nonlinear error evaluation of PGC demodulation, and displacement measurement were implemented to demonstrate the effectiveness of the proposed method. Experimental results show that the nonlinear error of phase demodulation reduced to less than ± 1° with real-time normalization, and nanometer displacement measurement is realized.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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2017 (4)

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

S. Zhang, L. Yan, B. Chen, Z. Xu, and J. Xie, “Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement,” Opt. Express 25(1), 472–485 (2017).
[Crossref] [PubMed]

A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. 17(13), 4143–4150 (2017).
[Crossref]

C. Ni, M. Zhang, Y. Zhu, C. Hu, S. Ding, and Z. Jia, “Sinusoidal phase-modulating interferometer with ellipse fitting and a correction method,” Appl. Opt. 56(13), 3895–3899 (2017).
[Crossref] [PubMed]

2016 (1)

2015 (2)

O. Sasaki, J. Xin, S. Choi, and T. Suzuki, “Profile measurement of thin films by backpropagation of multiple-wavelength optical fields with two sinusoidal phase-modulating interferometers,” Opt. Commun. 356, 578–581 (2015).
[Crossref]

W. Xia, Q. Liu, H. Hao, D. Guo, M. Wang, and X. Chen, “Sinusoidal phase-modulating self-mixing interferometer with nanometer resolution and improved measurement velocity range,” Appl. Opt. 54(26), 7820–7827 (2015).
[Crossref] [PubMed]

2013 (2)

H. Y. F. S. C. Huang and F. H. Hwang, “An improved sensitivity normalization technique of PGC demodulation with low minimum phase detection sensitivity using laser modulation to generate carrier signal,” Sensor Actuat. A-Phys. 191, 1–10 (2013).

K. Wang, M. Zhang, F. Duan, S. Xie, and Y. Liao, “Measurement of the phase shift between intensity and frequency modulations within DFB-LD and its influences on PGC demodulation in a fiber-optic sensor system,” Appl. Opt. 52(29), 7194–7199 (2013).
[Crossref] [PubMed]

2011 (1)

Q. Lin, L. Chen, S. Li, and X. Wu, “A high-resolution fiber optic accelerometer based on intracavity phase-generated carrier (PGC) modulation,” Meas. Sci. Technol. 22(1), 015303 (2011).
[Crossref]

2010 (1)

2007 (2)

S.-C. Huang and H. Lin, “Modified phase-generated carrier demodulation compensated for the propagation delay of the fiber,” Appl. Opt. 46(31), 7594–7603 (2007).
[Crossref] [PubMed]

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

2006 (1)

L. Wang, M. Zhang, X. Mao, and Y. Liao, “The arctangent approach of digital PGC demodulation for optic interferometric sensors,” Proc. SPIE 6292, 62921E (2006).
[Crossref]

2005 (1)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

2004 (1)

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

2000 (2)

T. Suzuki, K. Kobayashi, and O. Sasaki, “Real-time displacement measurement with a two-wavelength sinusoidal phase-modulating laser diode interferometer,” Appl. Opt. 39(16), 2646–2652 (2000).
[Crossref] [PubMed]

N. Servagent, T. Bosch, and M. Lescure, “Design of a phase-shifting optical feedback interferometer using an electrooptic modulator,” IEEE J. Sel. Top. Quantum Electron. 6(5), 798–802 (2000).

1997 (1)

T. Suzuki, “Real-time vibration measurement using a feedback type of laser diode interferometer with an optical fiber,” Opt. Eng. 36(9), 2496 (1997).
[Crossref]

1994 (1)

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).
[Crossref]

1991 (1)

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. 62(11), 2579–2583 (1991).
[Crossref]

1988 (1)

1982 (1)

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Aleynik, A. S.

A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. 17(13), 4143–4150 (2017).
[Crossref]

Bosch, T.

N. Servagent, T. Bosch, and M. Lescure, “Design of a phase-shifting optical feedback interferometer using an electrooptic modulator,” IEEE J. Sel. Top. Quantum Electron. 6(5), 798–802 (2000).

Chen, B.

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

S. Zhang, L. Yan, B. Chen, Z. Xu, and J. Xie, “Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement,” Opt. Express 25(1), 472–485 (2017).
[Crossref] [PubMed]

Chen, L.

Q. Lin, L. Chen, S. Li, and X. Wu, “A high-resolution fiber optic accelerometer based on intracavity phase-generated carrier (PGC) modulation,” Meas. Sci. Technol. 22(1), 015303 (2011).
[Crossref]

Chen, X.

Chen, Z.

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

Choi, S.

O. Sasaki, J. Xin, S. Choi, and T. Suzuki, “Profile measurement of thin films by backpropagation of multiple-wavelength optical fields with two sinusoidal phase-modulating interferometers,” Opt. Commun. 356, 578–581 (2015).
[Crossref]

Dandridge, A.

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Degl’Innocenti, R.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

Ding, S.

Docchio, F.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. 62(11), 2579–2583 (1991).
[Crossref]

Duan, F.

Gelmini, E.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. 62(11), 2579–2583 (1991).
[Crossref]

Giallorenzi, T.

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Guarino, A.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

Günter, P.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

Guo, D.

Hao, H.

He, J.

Hu, C.

Huang, H.

Huang, H. Y. F. S. C.

H. Y. F. S. C. Huang and F. H. Hwang, “An improved sensitivity normalization technique of PGC demodulation with low minimum phase detection sensitivity using laser modulation to generate carrier signal,” Sensor Actuat. A-Phys. 191, 1–10 (2013).

Huang, S.-C.

Hwang, F. H.

H. Y. F. S. C. Huang and F. H. Hwang, “An improved sensitivity normalization technique of PGC demodulation with low minimum phase detection sensitivity using laser modulation to generate carrier signal,” Sensor Actuat. A-Phys. 191, 1–10 (2013).

Jackson, D. A.

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).
[Crossref]

Jia, Z.

Kirkendall, C. K.

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

Kobayashi, K.

Lescure, M.

N. Servagent, T. Bosch, and M. Lescure, “Design of a phase-shifting optical feedback interferometer using an electrooptic modulator,” IEEE J. Sel. Top. Quantum Electron. 6(5), 798–802 (2000).

Li, F.

Li, Q.

Li, S.

Q. Lin, L. Chen, S. Li, and X. Wu, “A high-resolution fiber optic accelerometer based on intracavity phase-generated carrier (PGC) modulation,” Meas. Sci. Technol. 22(1), 015303 (2011).
[Crossref]

Liao, Y.

Lin, F.

Lin, H.

Lin, Q.

Q. Lin, L. Chen, S. Li, and X. Wu, “A high-resolution fiber optic accelerometer based on intracavity phase-generated carrier (PGC) modulation,” Meas. Sci. Technol. 22(1), 015303 (2011).
[Crossref]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Liu, Q.

Liu, Y.

Mao, X.

L. Wang, M. Zhang, X. Mao, and Y. Liao, “The arctangent approach of digital PGC demodulation for optic interferometric sensors,” Proc. SPIE 6292, 62921E (2006).
[Crossref]

Marioli, D.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. 62(11), 2579–2583 (1991).
[Crossref]

McGarrity, C.

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).
[Crossref]

Mekhrengin, M. V.

A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. 17(13), 4143–4150 (2017).
[Crossref]

Minoni, U.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. 62(11), 2579–2583 (1991).
[Crossref]

Miroshnichenko, G. P.

A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. 17(13), 4143–4150 (2017).
[Crossref]

Ni, C.

Plotnikov, M. Y.

A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. 17(13), 4143–4150 (2017).
[Crossref]

Poberaj, G.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Rezzonico, D.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

Sardini, E.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. 62(11), 2579–2583 (1991).
[Crossref]

Sasaki, O.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Servagent, N.

N. Servagent, T. Bosch, and M. Lescure, “Design of a phase-shifting optical feedback interferometer using an electrooptic modulator,” IEEE J. Sel. Top. Quantum Electron. 6(5), 798–802 (2000).

Suzuki, T.

O. Sasaki, J. Xin, S. Choi, and T. Suzuki, “Profile measurement of thin films by backpropagation of multiple-wavelength optical fields with two sinusoidal phase-modulating interferometers,” Opt. Commun. 356, 578–581 (2015).
[Crossref]

T. Suzuki, K. Kobayashi, and O. Sasaki, “Real-time displacement measurement with a two-wavelength sinusoidal phase-modulating laser diode interferometer,” Appl. Opt. 39(16), 2646–2652 (2000).
[Crossref] [PubMed]

T. Suzuki, “Real-time vibration measurement using a feedback type of laser diode interferometer with an optical fiber,” Opt. Eng. 36(9), 2496 (1997).
[Crossref]

Takahashi, K.

Tveten, A.

A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Volkov, A. V.

A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. 17(13), 4143–4150 (2017).
[Crossref]

Wang, K.

Wang, L.

J. He, L. Wang, F. Li, and Y. Liu, “An ameliorated phase generated carrier demodulation algorithm with low harmonic distortion and high stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).

L. Wang, M. Zhang, X. Mao, and Y. Liao, “The arctangent approach of digital PGC demodulation for optic interferometric sensors,” Proc. SPIE 6292, 62921E (2006).
[Crossref]

Wang, M.

Wu, X.

Q. Li, H. Huang, F. Lin, and X. Wu, “Optical micro-particle size detection by phase-generated carrier demodulation,” Opt. Express 24(11), 11458–11465 (2016).
[Crossref] [PubMed]

Q. Lin, L. Chen, S. Li, and X. Wu, “A high-resolution fiber optic accelerometer based on intracavity phase-generated carrier (PGC) modulation,” Meas. Sci. Technol. 22(1), 015303 (2011).
[Crossref]

Xia, W.

Xie, J.

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

S. Zhang, L. Yan, B. Chen, Z. Xu, and J. Xie, “Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement,” Opt. Express 25(1), 472–485 (2017).
[Crossref] [PubMed]

Xie, S.

Xin, J.

O. Sasaki, J. Xin, S. Choi, and T. Suzuki, “Profile measurement of thin films by backpropagation of multiple-wavelength optical fields with two sinusoidal phase-modulating interferometers,” Opt. Commun. 356, 578–581 (2015).
[Crossref]

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Xu, Z.

Yan, L.

S. Zhang, L. Yan, B. Chen, Z. Xu, and J. Xie, “Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement,” Opt. Express 25(1), 472–485 (2017).
[Crossref] [PubMed]

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

Zhang, E.

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

Zhang, M.

Zhang, S.

L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. 28(11), 115006 (2017).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic of PGC-Arctan demodulation with real-time normalization.
Fig. 2
Fig. 2 Schematic of the modified EOM-based SPMI.
Fig. 3
Fig. 3 Experimental setup of the modified EOM-based SPMI.
Fig. 4
Fig. 4 Results of real-time normalization verification. (a) Lissajous figure of P1(t) and P2(t). (b) Lissajous figure of Q1(t) and Q2(t). (c) Demodulated phases of the monitor and the probe interferometers. (d) Demodulated displacements of the monitor and the probe interferometers. The red dashed line is shifted by 50 nm to make the plots visible in (d).
Fig. 5
Fig. 5 Results of PGC demodulation before real-time normalization. (a) Demodulated phases. (b) Wrapped phase of φ2(t). (c) THD analysis of the wrapped phase of φ2(t).
Fig. 6
Fig. 6 Results of PGC demodulation after real-time normalization. (a) Demodulated phases. (b) Wrapped phase of φ2(t). (c) THD analysis of the wrapped phase of φ2(t).
Fig. 7
Fig. 7 Experimental results for nanometer displacement measurement. (a) Measurement results with the step of 20 nm in 5.12 μm. The red triangle line is shifted by 1 μm to make the plots visible. (b) FFT analysis of the displacement deviation.

Equations (17)

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V ω c ( t ) = A cos ω c t
S ( t ) = S 0 + S 1 cos [ z cos ( ω c t θ ) + φ ( t ) ]
S ( t ) = S 0 + S 1 cos φ ( t ) [ J 0 ( z ) + 2 m = 1 ( 1 ) m J 2 m ( z ) cos 2 m ( ω c t θ ) ] + S 1 sin φ ( t ) [ 2 m = 1 ( 1 ) m J 2 m 1 ( z ) cos ( 2 m 1 ) ( ω c t θ ) ] ,
P 1 ( t ) = L P F [ S ( t ) V ω c ( t ) ] J 1 ( z ) = J 1 ( z ) J 1 ( z ) K 1 S 1 A cos θ sin φ ( t )
P 2 ( t ) = L P F [ S ( t ) V 2 ω c ( t ) ] J 2 ( z ) = J 2 ( z ) J 2 ( z ) K 2 S 1 A cos 2 θ cos φ ( t )
Q 1 ( t ) = P 1 ( t ) α 1 = sin φ ( t )
Q 2 ( t ) = P 2 ( t ) α 2 = cos φ ( t )
φ ( t ) = arc tan Q 1 ( t ) Q 2 ( t )
V ( t ) = β V ω c ( t ) = β A cos ω c t
φ E O M = π β A V π cos ω c t
S 1 ( t ) = S 01 + S 11 cos { π β A V π cos ( ω c t θ 1 ) + 2 π λ [ l 1 + 2 l ( t ) ] } = S 01 + S 11 cos [ z cos ( ω c t θ 1 ) + φ 1 ( t ) ]
S 2 ( t ) = S 02 + S 12 cos { π β A V π cos ( ω c t θ 2 ) + 2 π λ [ l 2 + 2 l ( t ) + 2 d ( t ) ] } = S 02 + S 12 cos [ z cos ( ω c t θ 2 ) + φ 2 ( t ) ]
ϕ ( t ) = φ 2 ( t ) φ 1 ( t ) = 4 π λ d ( t ) + 2 π λ ( l 2 l 1 ) = 4 π λ d ( t ) + ϕ 0 .
Φ ( t ) = arc tan ( J 1 ( z ) J 2 ( z ) K 1 cos θ sin φ ( t ) J 1 ( z ) J 2 ( z ) K 2 cos 2 θ cos φ ( t ) ) = arc tan ( ν tan φ ( t ) )
Φ ( t ) = φ ( t ) + ν 1 2 sin 2 φ ( t )
Φ ( t ) = φ ( t ) + δ + ν 1 2 sin 2 [ φ ( t ) + δ ]
Δ Φ ( t ) = Φ ( t ) Φ ( t ) = δ + ν 1 2 { sin 2 [ φ ( t ) + δ ] sin 2 φ ( t ) } = δ + ( ν 1 ) sin δ cos [ 2 φ ( t ) + δ ]

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