Abstract

The first theory for two novel coherent beam combination architectures that are the first electronic beam combination architectures that completely eliminate the need for a separate reference beam are presented. Detailed theoretical models are developed and presented for the first time.

© 2006 Optical Society of America

Full Article  |  PDF Article
Related Articles
Method for analyzing multiple-mirror coupled optical systems

Jun Mizuno and Ichirou Yamaguchi
J. Opt. Soc. Am. A 16(7) 1730-1739 (1999)

First experimental demonstration of self-synchronous phase locking of an optical array

T. M. Shay, Vincent Benham, J. T. Baker, Benjamin Ward, Anthony D. Sanchez, Mark A. Culpepper, D. Pilkington, Justin Spring, Douglas J. Nelson, and Chunte A. Lu
Opt. Express 14(25) 12015-12021 (2006)

Coherent beam combination with single frequency dithering technique

Yanxing Ma, Pu Zhou, Xiaolin Wang, Haotong Ma, Xiaojun Xu, Lei Si, Zejin Liu, and Yijun Zhao
Opt. Lett. 35(9) 1308-1310 (2010)

References

  • View by:
  • |
  • |
  • |

  1. V. P. Gapontsev, “New milestones in the development of super high power fiber lasers,” presented at Photonics West, OE/LASE 2006, San Jose, CA, Jan 21–26, 2006.
  2. P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. 13, 439–441 (2001).
    [Crossref]
  3. E. J. Bochove, P. K. Cheo, and G. G. King, “Self-organization in a multicore fiber laser array,” Opt. Lett. 28, 1200–1202 (2003).
    [Crossref] [PubMed]
  4. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. I. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30, 1339–1341 (2003).
    [Crossref]
  5. R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
    [Crossref]
  6. R. A. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B 19, 1521–1534 (2002).
    [Crossref]
  7. C. J. Corcoran, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201121 (2005).
    [Crossref]
  8. B. W. Grimes, W. B. Roh, and T. G. Alley, “Beam phasing multiple fiber amplifiers using a fiber phase conjugate mirror,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61021C-1 to 61021C-8 (2006).
  9. R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).
  10. J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
    [Crossref]
  11. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of Yt fiber amplifiers,” Opt. Lett. 29, 474–476 (2004).
    [Crossref] [PubMed]
  12. M. Wickham, “Coherently coupled high power fiber arrays,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61020U-1 to 61020U-5 (2006).
  13. “A novel technique for phase locking Optical Fiber Arrays,” T. M. Shay and V. Benham, in Free-Space Laser Communications IV, J. C. Ricklin and D. G. Voelz, eds., Proc. SPIE5550, 313–319 (2004).
    [Crossref]
  14. “First experimental demonstration of fiber array phase locking by RF phase modulation,” T. M. Shay and V. Benham, Proceedings of the 17th Solid State and Diode Laser Technology Review, S. Ross, ed., pg. BEAM-7 (2004).
  15. “Self-synchronous locking of optical coherence by single-detector electronic-frequency tagging,” T. M. Shay, US Patent 7,058,098, June 2006.
  16. Note that in principle any odd harmonic of the modulation frequency, ωi, can be used to demodulate the phase error signal. However, the fundamental frequency generally produces the highest signal-to-noise ratio. Therefore, in this analysis the demodulation the fundamental frequency is always used for demodulation of the phase error signals.

2005 (1)

C. J. Corcoran, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201121 (2005).
[Crossref]

2004 (1)

2003 (2)

2002 (1)

2001 (1)

P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. 13, 439–441 (2001).
[Crossref]

Abderegg, J.

J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
[Crossref]

Alley, T. G.

B. W. Grimes, W. B. Roh, and T. G. Alley, “Beam phasing multiple fiber amplifiers using a fiber phase conjugate mirror,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61021C-1 to 61021C-8 (2006).

Alunni, D. A.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Augst, S. J.

Beach, R. A.

Beach, R. J.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Benham, V.

“First experimental demonstration of fiber array phase locking by RF phase modulation,” T. M. Shay and V. Benham, Proceedings of the 17th Solid State and Diode Laser Technology Review, S. Ross, ed., pg. BEAM-7 (2004).

“A novel technique for phase locking Optical Fiber Arrays,” T. M. Shay and V. Benham, in Free-Space Laser Communications IV, J. C. Ricklin and D. G. Voelz, eds., Proc. SPIE5550, 313–319 (2004).
[Crossref]

Bochove, E. J.

Brasure, L. D.

Brosnan, S. J.

J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
[Crossref]

Bruesselbach, H.

Cheo, P. K

P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. 13, 439–441 (2001).
[Crossref]

Cheo, P. K.

Corcoran, C. J.

C. J. Corcoran, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201121 (2005).
[Crossref]

Culter, K. P.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Davis, J. A.

R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

Dawson, J. W.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Fan, T. Y.

Feit, M. D.

R. A. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B 19, 1521–1534 (2002).
[Crossref]

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Gapontsev, V. P.

V. P. Gapontsev, “New milestones in the development of super high power fiber lasers,” presented at Photonics West, OE/LASE 2006, San Jose, CA, Jan 21–26, 2006.

Grimes, B. W.

B. W. Grimes, W. B. Roh, and T. G. Alley, “Beam phasing multiple fiber amplifiers using a fiber phase conjugate mirror,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61021C-1 to 61021C-8 (2006).

Hayden, J. S.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Hollister, J. H.

R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

Jones, D. C.

King, G. G.

E. J. Bochove, P. K. Cheo, and G. G. King, “Self-organization in a multicore fiber laser array,” Opt. Lett. 28, 1200–1202 (2003).
[Crossref] [PubMed]

P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. 13, 439–441 (2001).
[Crossref]

Komine, H.

J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
[Crossref]

Krashkevich, D.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Liu, A.

P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. 13, 439–441 (2001).
[Crossref]

Mangir, M. S.

Mead, R. W.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Minden, M. I.

Mitchell, S. C.

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Page, R. H.

Payne, S. A.

R. A. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B 19, 1521–1534 (2002).
[Crossref]

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

Rice et, R. R.

R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

Rogers, J. L.

Roh, W. B.

B. W. Grimes, W. B. Roh, and T. G. Alley, “Beam phasing multiple fiber amplifiers using a fiber phase conjugate mirror,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61021C-1 to 61021C-8 (2006).

Ruggieri, N. F.

R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

Sanchez, A.

Shay, T. M.

“A novel technique for phase locking Optical Fiber Arrays,” T. M. Shay and V. Benham, in Free-Space Laser Communications IV, J. C. Ricklin and D. G. Voelz, eds., Proc. SPIE5550, 313–319 (2004).
[Crossref]

“Self-synchronous locking of optical coherence by single-detector electronic-frequency tagging,” T. M. Shay, US Patent 7,058,098, June 2006.

“First experimental demonstration of fiber array phase locking by RF phase modulation,” T. M. Shay and V. Benham, Proceedings of the 17th Solid State and Diode Laser Technology Review, S. Ross, ed., pg. BEAM-7 (2004).

Weber, M. E.

J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
[Crossref]

Whitely, J. S.

R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

Wickham, M.

M. Wickham, “Coherently coupled high power fiber arrays,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61020U-1 to 61020U-5 (2006).

Wickham, M. G.

J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
[Crossref]

Wilcox, R.

Appl. Phys. Lett. (1)

C. J. Corcoran, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201121 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. K Cheo, A. Liu, and G. G. King, “A high brightness laser beam from a Phase-Locked Multicore Yb-Doped Fiber Laser Array,” IEEE Photon. Technol. Lett. 13, 439–441 (2001).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Lett. (3)

Other (10)

R. J. Beach, M. D. Feit, S. C. Mitchell, K. P. Culter, J. W. Dawson, S. A. Payne, R. W. Mead, J. S. Hayden, D. Krashkevich, and D. A. Alunni, “Ribbon fiber with multiple phase-locked gain cores,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 7–16 (2003).
[Crossref]

B. W. Grimes, W. B. Roh, and T. G. Alley, “Beam phasing multiple fiber amplifiers using a fiber phase conjugate mirror,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61021C-1 to 61021C-8 (2006).

R. R. Rice et, J. A. Davis, J. S. Whitely, J. H. Hollister, and N. F. Ruggieri, “Coherent Fiber MOPA,” Presented at 14th Annual Solid State and Diode Laser Technology Review, Sean Ross, ed., Albuquerque, NM (2001).

J. Abderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-watt coherently-phased 4-element fiber array,” in Advances in Fiber Lasers, L. N. Durvasula, ed., Proc. SPIE4974, 1–6 (2003).
[Crossref]

M. Wickham, “Coherently coupled high power fiber arrays,” in Fiber Lasers III: Technology, Systems, and Applications, A. J. W. Brown, J. Nilsson, D. J. Harter, and A. Tunnermann, eds., Proc. SPIE6102, 61020U-1 to 61020U-5 (2006).

“A novel technique for phase locking Optical Fiber Arrays,” T. M. Shay and V. Benham, in Free-Space Laser Communications IV, J. C. Ricklin and D. G. Voelz, eds., Proc. SPIE5550, 313–319 (2004).
[Crossref]

“First experimental demonstration of fiber array phase locking by RF phase modulation,” T. M. Shay and V. Benham, Proceedings of the 17th Solid State and Diode Laser Technology Review, S. Ross, ed., pg. BEAM-7 (2004).

“Self-synchronous locking of optical coherence by single-detector electronic-frequency tagging,” T. M. Shay, US Patent 7,058,098, June 2006.

Note that in principle any odd harmonic of the modulation frequency, ωi, can be used to demodulate the phase error signal. However, the fundamental frequency generally produces the highest signal-to-noise ratio. Therefore, in this analysis the demodulation the fundamental frequency is always used for demodulation of the phase error signals.

V. P. Gapontsev, “New milestones in the development of super high power fiber lasers,” presented at Photonics West, OE/LASE 2006, San Jose, CA, Jan 21–26, 2006.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1.
Fig. 1.

A linear control loop model for self-synchronous LOCSET.

Equations (27)

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

E u ( t ) = E u 0 · Cos ( ω L · t + ϕ u ) and
E i ( t ) = E i 0 · Cos ( ω L · t + ϕ i + β i · Sin ( ω i · t ) ) ,
E i ( t ) = E i 0 · [ Cos ( ω L · t + ϕ i ) Cos ( β i · Sin ( ω i · t ) ) Sin ( ω L · t + ϕ i ) Sin ( β i · Sin ( ω i · t ) ) ] ,
i PD ( t ) = R PD · A · μ o ε o · { E u 2 ( t ) + ( l = 1 N E l ( t ) ) ( j = 1 N E j ( t ) ) + 2 · E u ( t ) · j = 1 N E j ( t ) } ,
i u 2 ( t ) = ε o μ o · R PD · A · E u 2 ( t ) ,
i uj ( t ) = ε o μ o · R PD · A · 2 · E u ( t ) · j = 1 N E j ( t ) .
i ij ( t ) = ε o μ o · R PD · A · ( l = 1 N E l ( t ) ) ( j = 1 , j l N E j ( t ) ) .
i u 2 ( t ) = R PD · P u 2 .
i lj ( t ) = R PD 2
l = 1 N P l · j = 1 j l N P j [ { Cos ( ϕ l ϕ j ) ( J 0 ( β l ) + 2 · n l = 1 J 2 n l ( β l ) · Cos ( 2 · n l · ω l · t ) ) ( J 0 ( β j ) + 2 · n j = 1 J 2 n j ( β j ) · Cos ( 2 · n j · ω j · t ) ) } { Sin ( ϕ l ϕ j ) · 2 · n l = 1 J 2 n l 1 ( β l ) · Sin ( ( 2 · n l 1 ) · ω l · t ) ) ( J 0 ( β j ) + 2 · n j = 1 J 2 n j ( β j ) · Cos ( 2 · n j · ω j · t ) ) } + { Sin ( ϕ l ϕ j ) ( J 0 ( β l ) + 2 · n j = 1 J 2 n j ( β j ) · Cos ( 2 · n j · ω j · t ) ) 2 · n j = 1 J 2 n j 1 ( β j ) · Sin ( ( 2 · n j 1 ) · ω j · t ) } + { Cos ( ϕ l ϕ j ) ( 2 · n j = 1 J 2 n j 1 ( β j ) ·Sin ( ( 2 · n j 1 ) · ω j · t ) ) 2 · n l = 1 J 2 n l 1 ( β l ) ·Sin ( ( 2 · n l 1 ) · ω l · t ) } ] ,
i uj ( t ) = R PD · P u ·
j = 1 N P j · [ { Cos ( ϕ j ϕ u ) ( J 0 ( β j ) + 2 · n j = 1 J 2 n j ( β j ) · Cos ( 2 · n j · ω j · t ) ) } + { Sin ( ϕ j ϕ u ) · 2 · n j = 1 J 2 n j 1 ( β j ) · Sin ( ( 2 · n j 1 ) · ω j · t ) } ] ,
S uic = 1 τ · 0 τ i ui ( t ) · Sin ( ω c · t ) · dt =
1 τ · 0 τ { 2 · R PD · P u · Sin ( ω c · t ) i = 1 N P i · [ { Cos ( ϕ i ϕ u ) ( J 0 ( β i ) + 2 · n i = 1 J 2 n i ( β i ) · Cos ( 2 · n i · ω i · t ) ) } + { Sin ( ϕ u ϕ i ) · 2 · n i = 1 J 2 n i 1 ( β i ) · Sin ( ( 2 · n i 1 ) · ω i · t ) } ] } · dt .
S uii = R PD · P u · P i · Sin ( ϕ u ϕ i ) · J 1 ( β i ) ,
S ijc = 1 τ 0 τ i ij ( t ) · Sin ( ω c · t ) · dt =
S iji = R PD · p i · J 1 ( β i ) · j = 1 N p j · J 0 ( β j ) · Sin ( ϕ j ϕ i ) ,
S SRi = S uii + S iji
S SRi = R PD · P i · J 1 ( β i ) ( P u · Sin ( ϕ u ϕ i ) + j = 1 N P j · J 0 ( β j ) · Sin ( ϕ j ϕ i ) ) ,
S SSi = S iji = R PD · P i · J 1 ( β i ) ( j = 1 N P j · J 0 ( β j ) · Sin ( ϕ j ϕ i ) ) ,
ϕ j ( t ) = ϕ jo + Δ ϕ j ( t ) ,
S SSi = R PD · P i · J 1 ( β i ) ( j = 1 N P j · J 0 ( β j ) · Sin ( ϕ jo ϕ io + δ ϕ j ( t ) Δ ϕ i ( t ) ) ) ,
S SSi = R PD · P i · J 1 ( β i ) ( j = 1 N P j · J 0 ( β j ) · [ δ ϕ je ( t ) Δ ϕ i ( t ) ] ) .
K ij = R PD · P i · J 1 ( β i ) · P j · J 0 ( β j ) .
δ ϕ ie _ i ( s ) = s · Δ ϕ i s + A e · K PM τ · j = 1 j i N K ij ,
SNR i = ( δ ϕ ie _ rms · R PD · J 1 ( β i ) · P i · j = 1 j 1 N J 0 ( β j ) · P j ) 2 2 · q · B · R PD · j = 1 N P j ,
SNR i δ ϕ ie _ rms 2 · R PD · J 1 ( β i ) 2 · P i · π · τ q · A e · K PM · K ij ,

Metrics