Abstract

In this manuscript, the influence of perturbative phase noise on active coherent polarization beam combining (CPBC) system is studied theoretically and experimentally. By employing a photo-detector to obtain phase error signal for feedback loop, actively coherent polarization beam combining of two 20 W-level single mode polarization-maintained (PM) fiber amplifiers are demonstrated with more than 94% combining efficiency. Then the influence of perturbative phase noise on active CPBC system is illustrated by incorporating a simulated phase noise signal in one of the two amplifiers. Experimental results show that the combining efficiency of the CPBC system is susceptible to the frequency or amplitude of the perturbative phase noise. In order to ensure the combining efficiency of the unit of CPBC system higher than 90%, the competence of our active phase control module for high power operation is discussed, which suggests that it could be worked at 100s W power level. The relationship between residual phase noise of the active controller and the normalized voltage signal of the photo-detector is developed and validated experimentally. Experimental results correspond exactly with the theoretically analyzed combining efficiency. Our method offers a useful approach to estimate the influence of phase noise on CPBC system.

© 2013 Optical Society of America

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  1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
    [Crossref] [PubMed]
  2. Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
    [Crossref]
  3. J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19(18), 17053–17058 (2011).
    [Crossref] [PubMed]
  4. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
    [Crossref] [PubMed]
  5. P. Zhou, Z. J. Liu, X. J. Xu, and Z. L. Chen, “Numerical analysis of the effects of aberrations on coherently combined fiber laser beams,” Appl. Opt. 47(18), 3350–3359 (2008).
    [Crossref] [PubMed]
  6. G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
    [Crossref]
  7. T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006).
    [Crossref] [PubMed]
  8. P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
    [Crossref]
  9. X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
    [Crossref] [PubMed]
  10. C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
    [Crossref]
  11. E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
    [Crossref]
  12. J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
    [Crossref] [PubMed]
  13. P. B. Phua and Y. L. Lim, “Coherent polarization locking with near-perfect combining efficiency,” Opt. Lett. 31(14), 2148–2150 (2006).
    [Crossref] [PubMed]
  14. L. H. Tan, C. F. Chua, and P. B. Phua, “Preserving a diffraction-limited beam in Ho:YAG laser using coherent polarization locking,” Opt. Lett. 37(22), 4621–4623 (2012).
    [Crossref] [PubMed]
  15. R. Uberna, A. Bratcher, and B. G. Tiemann, “Power scaling of a fiber master oscillator power amplifier system using a coherent polarization beam combination,” Appl. Opt. 49(35), 6762–6765 (2010).
    [Crossref] [PubMed]
  16. R. Uberna, A. Bratcher, and B. G. Tiemann, “Power scaling of a fiber master oscillator power amplifier system using a coherent polarization beam combination,” Appl. Opt. 49(35), 6762–6765 (2010).
    [Crossref] [PubMed]
  17. P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
    [Crossref]
  18. P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
    [Crossref]
  19. D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007).
    [Crossref] [PubMed]
  20. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004).
    [Crossref] [PubMed]
  21. H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012).
    [Crossref] [PubMed]
  22. M. Tröbs, S. Barke, T. Theeg, D. Kracht, G. Heinzel, and K. Danzmann, “Differential phase-noise properties of a ytterbium-doped fiber amplifier for the Laser Interferometer Space Antenna,” Opt. Lett. 35(3), 435–437 (2010).
    [Crossref] [PubMed]
  23. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
    [Crossref] [PubMed]
  24. Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
    [Crossref]
  25. S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb³⁺-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011).
    [Crossref] [PubMed]
  26. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
    [Crossref] [PubMed]

2013 (1)

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

2012 (4)

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012).
[Crossref] [PubMed]

L. H. Tan, C. F. Chua, and P. B. Phua, “Preserving a diffraction-limited beam in Ho:YAG laser using coherent polarization locking,” Opt. Lett. 37(22), 4621–4623 (2012).
[Crossref] [PubMed]

2011 (5)

2010 (4)

2009 (2)

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

2008 (2)

2007 (4)

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007).
[Crossref] [PubMed]

J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (1)

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

2004 (1)

Anderegg, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Asman, C. P.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Augst, S. J.

Baker, J. T.

Barke, S.

Barthelemy, A.

Barty, C. P. J.

Beach, R. J.

Bellanger, C.

Benham, V.

Bochove, E. J.

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

Bourderionnet, J.

Bratcher, A.

Brignon, A.

Brosnan, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Chen, D. D.

Chen, Z. L.

Cheung, E. C.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Chua, C. F.

Corcoran, C. J.

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

Culpepper, M. A.

Danzmann, K.

Dawson, J. W.

Desfarges-Berthelemot, A.

Dong, X.

Du, W.

Durville, F.

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

Fan, T. Y.

Feng, Y.

Goldizen, K. C.

Goodno, G. D.

G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
[Crossref] [PubMed]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Guo, S.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Hammons, D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

He, B.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Heebner, J. E.

Heinzel, G.

Hickey, L. M. B.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Horley, R.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Injeyan, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Jeong, Y.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Jones, D. C.

Kermene, V.

Komine, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Kracht, D.

Leng, J.

Leng, J. Y.

Lhermite, J.

Li, Z.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Lim, Y. L.

Liu, C.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Liu, Z.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Liu, Z. J.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

P. Zhou, Z. J. Liu, X. J. Xu, and Z. L. Chen, “Numerical analysis of the effects of aberrations on coherently combined fiber laser beams,” Appl. Opt. 47(18), 3350–3359 (2008).
[Crossref] [PubMed]

Long, W. H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Lou, Q.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Lu, C. A.

Ma, H.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Ma, P. F.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Ma, Y.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Ma, Y. X.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

McClellan, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

McNaught, S. J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Messerly, M. J.

Murphy, D. V.

Nelson, D. J.

Neumann, J.

Nilsson, J.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Pax, P. H.

Payne, D. N.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Peng, M. Y.

Phua, P. B.

Pilkington, D.

Primot, J.

Qi, Y.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Qian, Q.

Qiu, J. R.

Redmond, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Redmond, S. M.

Rothenberg, J. E.

Sahu, J. K.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Sanchez, A.

Sanchez, A. D.

Scott, A. M.

Shakir, S. A.

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

Shay, T. M.

Shen, S. X.

Shih, C. C.

Shverdin, M. Y.

Si, L.

Siders, C. W.

Simpson, R.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Sollee, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Spring, J.

Sridharan, A. K.

Stacey, C. D.

Stappaerts, E. A.

Su, R. T.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Tan, L. H.

Theeg, T.

Tiemann, B. G.

Tröbs, M.

Tünnermann, H.

Turner, P. W.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

Uberna, R.

Wang, X.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Wang, X. L.

Ward, B.

Weber, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Wei, X. M.

Weiss, S. B.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Weßels, P.

Wickham, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Xiao, H.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

Xu, S. H.

Xu, X.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

Xu, X. J.

Xue, Y.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Yang, Z. M.

Yu, C. X.

Zhang, Q. Y.

Zhang, W. N.

Zhao, Y.

Zhou, J.

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

Zhou, P.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

P. Zhou, Z. J. Liu, X. J. Xu, and Z. L. Chen, “Numerical analysis of the effects of aberrations on coherently combined fiber laser beams,” Appl. Opt. 47(18), 3350–3359 (2008).
[Crossref] [PubMed]

Zhu, J.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase—locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86(20), 201118 (2005).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (4)

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15(2), 320–327 (2009).
[Crossref]

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007).
[Crossref]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solidstate laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combination of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15(2), 248–256 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (2)

Z. Li, J. Zhou, B. He, Y. Xue, P. Zhou, C. Liu, Y. Qi, Q. Lou, and X. Xu, “Impact of phase perturbation on passive phase-locking coherent beam combination,” IEEE Photon. Technol. Lett. 24(8), 655–657 (2012).
[Crossref]

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

Laser Phys. Lett. (1)

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Opt. Express (4)

Opt. Lett. (11)

S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb³⁺-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011).
[Crossref] [PubMed]

H. Tünnermann, Y. Feng, J. Neumann, D. Kracht, and P. Weßels, “All-fiber coherent beam combining with phase stabilization via differential pump power control,” Opt. Lett. 37(7), 1202–1204 (2012).
[Crossref] [PubMed]

L. H. Tan, C. F. Chua, and P. B. Phua, “Preserving a diffraction-limited beam in Ho:YAG laser using coherent polarization locking,” Opt. Lett. 37(22), 4621–4623 (2012).
[Crossref] [PubMed]

D. C. Jones, C. D. Stacey, and A. M. Scott, “Phase stabilization of a large-mode-area ytterbium-doped fiber amplifier,” Opt. Lett. 32(5), 466–468 (2007).
[Crossref] [PubMed]

J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
[Crossref] [PubMed]

M. Tröbs, S. Barke, T. Theeg, D. Kracht, G. Heinzel, and K. Danzmann, “Differential phase-noise properties of a ytterbium-doped fiber amplifier for the Laser Interferometer Space Antenna,” Opt. Lett. 35(3), 435–437 (2010).
[Crossref] [PubMed]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
[Crossref] [PubMed]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004).
[Crossref] [PubMed]

P. B. Phua and Y. L. Lim, “Coherent polarization locking with near-perfect combining efficiency,” Opt. Lett. 31(14), 2148–2150 (2006).
[Crossref] [PubMed]

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Figures (13)

Fig. 1
Fig. 1 The polarization state of combined beam for (a) without phase control and (b) phase locked.

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Fig. 2
Fig. 2 Experimental setup of the two-channel CPBC system with artificial phase noise. PM: phase modulator; CO1-CO2: collimators; HWP: half wavelength plate; M1-M2: all- reflected mirrors; M3: high-reflected mirror (99.9:0.1); M4:96:4 splitter; PBC1-PBC2: polarization beam combiners; P: linear polarizer; PD-photo-detector. P1: Power meter 1; P2: Power meter 2.

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Fig. 3
Fig. 3 The principle diagram of the experimental setup.

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Fig. 4
Fig. 4 The spectral density of power of phase noise in the two amplifier chains. (a) First amplifier in 3 W level. (b) First amplifier in 20 W level. (c) Second amplifier in 3 W level. (d) Second amplifier in 20 W level.

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Fig. 5
Fig. 5 Time series signals and spectral density of energy encircled in the pinhole in open loop and closed loop. (a) Time series signals. (b) Spectral density.

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Fig. 6
Fig. 6 The intensity profiles with the system in open loop and closed loop. (a) Three intensity profiles in open loop. (b) Intensity profiles in closed loop with phase locking to minimum (left) and maximum (right) state.

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Fig. 7
Fig. 7 The change of normalized voltage signals along with the modulation depth with the modulation frequency of 200 Hz.

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Fig. 8
Fig. 8 The change of normalized voltage signals along with the modulation depth with the modulation frequency of 400 Hz.

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Fig. 9
Fig. 9 The contrast of estimated combining efficiency and the measured combining efficiency. (a) Modulation frequency: 200 Hz. (b) Modulation frequency: 400 Hz.

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Fig. 10
Fig. 10 The change of normalized voltage signals along with the modulation frequency (f) with the modulation depth of λ/4.

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Fig. 11
Fig. 11 The change of normalized voltage signals along with the modulation frequency (f) with the modulation depth of λ/2.

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Fig. 12
Fig. 12 The contrast of estimated combining efficiency and the measured combining efficiency. (a) Modulation depth: λ/4. (b) Modulation depth: λ/2.

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Fig. 13
Fig. 13 (a) The trend of combining efficiency along with modulation frequency for different phase modulation depth. (b) The required finitude band frequency (fr) of the phase noise of the amplifier chain in the circumstance of different modulation depth.

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Equations (12)

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

E 1 (x,y)= ψ 1 (x,y)exp(i ϕ 1 ) E 2 (x,y)= ψ 2 (x,y)exp(i ϕ 2 )
ψ 1 (x,y)= 2 P b1 π . 1 w 1 .exp[ − ( x 2 + y 2 ) w 1 2 ] ψ 2 (x,y)= 2 P b2 π . 1 w 2 .exp[ − ( x 2 + y 2 ) w 2 2 ]
P b1 = ∬ | E 1 (x,y) | 2 dxdy P b2 = ∬ | E 2 (x,y) | 2 dxdy
η= < P c (t) > t P 0
< P c (t)>= 1 2 ∫ −∞ ∞ ∫ −∞ ∞ [ ψ 2 1 (x,y)+ ψ 2 2 (x,y) ] d x d y + 1 2T ∫ 0 T 4 ξ 2 1 + ξ 2 2 d t
ξ 1 (t)= ∫ −∞ ∞ ∫ −∞ ∞ ψ (x,y) 1 ψ (x,y) 2 cosδ(t) d x d y ξ 2 = ∫ −∞ ∞ ∫ −∞ ∞ [ ψ 2 1 (x,y)− ψ 2 2 (x,y) ] d x d y
P 0 = ∫ −∞ ∞ ∫ −∞ ∞ [ ψ 2 1 (x,y)+ ψ 2 2 (x,y) ] d x d y
cosδ(t)= V(t)− κ 1 cos 2 θ− κ 2 sin 2 θ 2 κ 1 κ 2 sin(θ)cos(θ)
κ 1 = P b1 P 0 κ 2 = P b2 P 0
η={ 1 T ∫ 0 T V(t) d t phase locking at maximum state 1 T ∫ 0 T [ 1−V(t) ] d t phase locking at minimum state
η total =η− η 0
η={ 1 N ∑ i=1 N V(i) phase locking at maximum state 1 N ∑ i=1 N [ 1−V(i) ] phase locking at minimum state

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