Abstract

A method for simultaneously calibrating the peak retardation and static retardation of a photoelastic modulator (PEM) is proposed. By optimizing the polarization modulation system, the normalized fundamental frequency components of the modulation signals are obtained to calculate the peak retardation and static retardation of the PEM. The calibration result is immune to fluctuations of the incident light intensity and can be used to correct the deviation of the photoelastic modulation detection results. In our experiments, the average deviation between measured peak retardations and corresponding set values is 0.0339 rad. The standard deviation between the measured static retardations and the average measured value is 0.0003 rad. The calibration method has a high sensitivity, since the large gradient of the first-order Bessel function when the peak retardation is less than 1 rad. The experimental results are consistent with the theoretical analysis.

© 2017 Optical Society of America

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References

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  1. J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
    [Crossref]
  2. J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
    [Crossref]
  3. J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
    [Crossref]
  4. J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
    [Crossref]
  5. W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015).
    [Crossref]
  6. J. M. Brown, “A new limit on Lorentz-and CPT-violating neutron spin interactions using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2011).
  7. L. H. Duan, J. C. Fang, R. J. Li, L. W. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express 23, 32481–32489 (2015).
    [Crossref]
  8. A. J. Zeng, F. Y. Li, L. L. Zhu, and H. J. Huang, “Simultaneous measurement of retardance and fast axis angle of a quarter-wave plate using one photoelastic modulator,” Appl. Opt. 50, 4347–4352 (2011).
    [Crossref]
  9. B. Wang and J. List, “Basic optical properties of the photoelastic modulator: Part I. Useful aperture and acceptance angle,” Proc. SPIE 5888, 58881I (2005).
    [Crossref]
  10. B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009).
    [Crossref]
  11. M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
    [Crossref]
  12. M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004).
    [Crossref]
  13. S. A. Wan, “Study on experiment of error analysis and suppression methods for SERF atomic spin gyroscope,” Ph.D. dissertation (Beihang University, 2014).
  14. T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000).
    [Crossref]
  15. K. Li, “High sensitive measurement of optical rotation based on photo-elastic modulation,” Acta Phys. Sinica 64, 5837–5843 (2015).
    [Crossref]
  16. R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993).
    [Crossref]
  17. Y. Zhang, F. J. Song, H. Y. Li, and X. G. Yang, “Precise measurement of optical phase retardation of a wave plate using modulated-polarized light,” Appl. Opt. 49, 5837–5843 (2010).
    [Crossref]

2015 (4)

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015).
[Crossref]

L. H. Duan, J. C. Fang, R. J. Li, L. W. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express 23, 32481–32489 (2015).
[Crossref]

K. Li, “High sensitive measurement of optical rotation based on photo-elastic modulation,” Acta Phys. Sinica 64, 5837–5843 (2015).
[Crossref]

2014 (2)

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

2013 (1)

J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009).
[Crossref]

2005 (1)

B. Wang and J. List, “Basic optical properties of the photoelastic modulator: Part I. Useful aperture and acceptance angle,” Proc. SPIE 5888, 58881I (2005).
[Crossref]

2004 (2)

M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
[Crossref]

M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004).
[Crossref]

2000 (1)

T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000).
[Crossref]

1993 (1)

R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993).
[Crossref]

Brown, J. M.

J. M. Brown, “A new limit on Lorentz-and CPT-violating neutron spin interactions using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2011).

Cai, H. W.

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

Chao, Y. F.

M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
[Crossref]

M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004).
[Crossref]

Cline, R. A.

R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993).
[Crossref]

Ding, M.

Duan, L. H.

Fang, J. C.

L. H. Duan, J. C. Fang, R. J. Li, L. W. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express 23, 32481–32489 (2015).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
[Crossref]

Hinds, E.

B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009).
[Crossref]

Huang, H. J.

Jiang, L. W.

Krivoy, E.

B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009).
[Crossref]

Leou, K. C.

M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
[Crossref]

Li, F. Y.

Li, H. Y.

Li, K.

K. Li, “High sensitive measurement of optical rotation based on photo-elastic modulation,” Acta Phys. Sinica 64, 5837–5843 (2015).
[Crossref]

Li, R. J.

Li, Y.

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

List, J.

B. Wang and J. List, “Basic optical properties of the photoelastic modulator: Part I. Useful aperture and acceptance angle,” Proc. SPIE 5888, 58881I (2005).
[Crossref]

Liu, B. Q.

W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015).
[Crossref]

Oakberg, T. C.

T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000).
[Crossref]

Qin, J.

J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
[Crossref]

Quan, W.

W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015).
[Crossref]

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

Risley, J. S.

R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993).
[Crossref]

Song, F. J.

Sutherland, J. C.

T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000).
[Crossref]

Trunk, J.

T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000).
[Crossref]

Tsai, F. H.

M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004).
[Crossref]

M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
[Crossref]

Wan, S. A.

J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
[Crossref]

S. A. Wan, “Study on experiment of error analysis and suppression methods for SERF atomic spin gyroscope,” Ph.D. dissertation (Beihang University, 2014).

Wang, B.

B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009).
[Crossref]

B. Wang and J. List, “Basic optical properties of the photoelastic modulator: Part I. Useful aperture and acceptance angle,” Proc. SPIE 5888, 58881I (2005).
[Crossref]

Wang, M. W.

M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
[Crossref]

M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004).
[Crossref]

Wang, T.

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

Wang, W.

Westerveld, W. B.

R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993).
[Crossref]

Yang, X. G.

Yuan, H.

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

Zeng, A. J.

Zhang, H.

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

Zhang, Y.

Zhu, L. L.

Zou, S.

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

Acta Phys. Sinica (1)

K. Li, “High sensitive measurement of optical rotation based on photo-elastic modulation,” Acta Phys. Sinica 64, 5837–5843 (2015).
[Crossref]

Appl. Opt. (2)

Chin. Phys. B (1)

J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015).
[Crossref]

Chin. Sci. Bull. (1)

J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
[Crossref]

Europhys. Lett. (1)

W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015).
[Crossref]

Jpn. J. Appl. Phys. (1)

M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004).
[Crossref]

Opt. Express (1)

Proc. SPIE (3)

T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000).
[Crossref]

B. Wang and J. List, “Basic optical properties of the photoelastic modulator: Part I. Useful aperture and acceptance angle,” Proc. SPIE 5888, 58881I (2005).
[Crossref]

B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009).
[Crossref]

Rev. Sci. Instrum. (3)

J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014).
[Crossref]

J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014).
[Crossref]

R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993).
[Crossref]

Thin Solid Films (1)

M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004).
[Crossref]

Other (2)

S. A. Wan, “Study on experiment of error analysis and suppression methods for SERF atomic spin gyroscope,” Ph.D. dissertation (Beihang University, 2014).

J. M. Brown, “A new limit on Lorentz-and CPT-violating neutron spin interactions using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2011).

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Figures (4)

Fig. 1.
Fig. 1. Plots of Bessel functions versus the peak retardation.

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Fig. 2.
Fig. 2. Schematic for the calibration of the PEM.

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Fig. 3.
Fig. 3. Calibration results of peak retardation.

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Fig. 4.
Fig. 4. Calibration results of static retardation.

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Equations (36)

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

G 0 = E 0 1 2 [ 1 1 ] ,
G λ / 4 = [ 1 0 0 i ] .
G PEM = [ cos   δ ( t ) 2 − i   sin   δ ( t ) 2 0 0 cos   δ ( t ) 2 + i   sin   δ ( t ) 2 ] ,
δ ( t ) = δ 0   sin ( ω t ) + δ s ,
G λ / 2 = ( − i 2 ) [ 1 1 1 − 1 ] .
G PBS 1 = [ 0 0 0 1 ] ,
G PBS 2 = [ 1 0 0 0 ] .
G 1 = G PBS 1 G λ / 2 G PEM G λ / 4 G 0 ,
G 2 = G PBS 2 G λ / 2 G PEM G λ / 4 G 0 .
I 1 = G 1 * G 1 = I 0 2 [ 1 + sin ( δ 0   sin   ω t + δ s ) ] = I 0 2 [ 1 + sin ( δ 0   sin   ω t ) cos   δ s + cos ( δ 0   sin   ω t ) sin   δ s ] ,
I 2 = G 2 * G 2 = I 0 2 [ 1 − sin ( δ 0   sin   ω t + δ s ) ] = I 0 2 [ 1 − sin ( δ 0   sin   ω t ) cos   δ s − cos ( δ 0   sin   ω t ) sin   δ s ] ,
sin ( δ 0   sin   ω t ) = 2 ∑ k = 1 ∞ J 2 k − 1 ( δ 0 ) sin [ ( 2 k − 1 ) ω t ] ,
cos ( δ 0   sin   ω t ) = J 0 ( δ 0 ) + 2 ∑ k = 1 ∞ J 2 k ( δ 0 ) cos [ ( 2 k ) ω t ] ,
I 1 = I 0 2 [ 1 + J 0 ( δ 0 ) sin   δ s + 2 J 1 ( δ 0 ) cos   δ s   sin ( ω t ) + 2 J 2 ( δ 0 ) sin   δ s   cos ( 2 ω t ) + … ] ,
I 2 = I 0 2 [ 1 − J 0 ( δ 0 ) sin   δ s − 2 J 1 ( δ 0 ) cos   δ s   sin ( ω t ) − 2 J 2 ( δ 0 ) sin   δ s   cos ( 2 ω t ) − … ] .
V dc 1 = η I 0 2 [ 1 + J 0 ( δ 0 ) sin   δ s ] ,
V dc 2 = η I 0 2 [ 1 − J 0 ( δ 0 ) sin   δ s ] ,
V 1 f 1 = η I 0 J 1 ( δ 0 ) cos   δ s ,
V 1 f 2 = η I 0 J 1 ( δ 0 ) cos   δ s .
W = V 1 f 1 + V 1 f 2 V dc 1 + V dc 2 = 2 J 1 ( δ 0 ) cos   δ s .
G 1 ′ = G PBS 1 G λ / 2 G PEM G 0 ,
G 2 ′ = G PBS 2 G λ / 2 G PEM G 0 .
I 1 ′ = G 1 ′ * G 1 ′ = I 0 2 [ 1 − cos   δ ( t ) ] = I 0 2 [ 1 − cos ( δ 0   sin   ω t + δ s ) ] ,
I 2 ′ = G 2 ′ * G 2 ′ = I 0 2 [ 1 + cos   δ ( t ) ] = I 0 2 [ 1 + cos ( δ 0   sin   ω t + δ s ) ] .
I 1 ′ = I 0 2 [ 1 − J 0 ( δ 0 ) cos   δ s + 2 J 1 ( δ 0 ) sin   δ s   sin ( ω t ) − 2 J 2 ( δ 0 ) cos   δ s   cos ( 2 ω t ) + … ] ,
I 2 ′ = I 0 2 [ 1 + J 0 ( δ 0 ) cos   δ s − 2 J 1 ( δ 0 ) sin   δ s   sin ( ω t ) + 2 J 2 ( δ 0 ) cos   δ s   cos ( 2 ω t ) + … ] .
V dc 1 ′ = η I 0 2 [ 1 − J 0 ( δ 0 ) cos   δ s ] ,
V dc 2 ′ = η I 0 2 [ 1 + J 0 ( δ 0 ) cos   δ s ] ,
V 1 f 1 ′ = η I 0 J 1 ( δ 0 ) sin   δ s ,
V 1 f 2 ′ = η I 0 J 1 ( δ 0 ) sin   δ s .
W ′ = V 1 f 1 ′ + V 1 f 2 ′ V dc 1 ′ + V dc 2 ′ = 2 J 1 ( δ 0 ) sin   δ s .
J 1 ( δ 0 ) = W 2 + W ′ 2 2 ,
δ s = tan − 1 W ′ W .
J 1 ( δ 0 ) = 0.5787   sin ( 0.8638 δ 0 + 0.000026 ) .
δ 0 = [ sin − 1 ( W 2 + W ′ 2 / 1.1574 ) − 0.000026 ] / 0.8638 .
δ 0 M = 0.9983 δ 0 set + 0.0339 ,

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