Abstract

The array of two mutually-injected fiber lasers is theoretically studied. It is found that the self-organization mechanism of this array is virtually the longitudinal-mode competition in the compound laser cavity. Two phase-locked states of this array are predicted. The performance of this array is also investigated, and the theoretical result agrees well with the experimental observation. Based on the theoretical analysis, some advices for building this array are also given.

© 2009 Optical Society of America

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References

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  1. T. Y. Fan, "Laser beam combining for high-power, high-radiance sources," IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005).
    [CrossRef]
  2. Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. 42, 26-29 (2005). (in Chinese)
  3. J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)
  4. J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
    [CrossRef]
  5. T. M. Shay, "Theory of electronically phased coherent beam combination without a reference beam," Opt. Express 14, 12189-12195 (2006), http://www.opticsinfobase.org/abstract.cfm?&uri=oe-14-25-12188.
    [CrossRef]
  6. 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, 12015-12021 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-14-25-12015.
    [CrossRef] [PubMed]
  7. J. Morel, A. Woodtli, and R. Daendliker, "Coherent coupling of an array of Nd3+ doped single-mode fiber lasers using an intracavity phase grating," Proc. SPIE 1789,13-17 (1992).
    [CrossRef]
  8. M. Wrage, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, "Phase locking in a multicore fiber laser by means of a Talbot resonator," Opt. Lett. 25, 1436-1438 (2000).
    [CrossRef]
  9. L. Li, A. Schulzgen, S. Chen, and V. L. Temyanko, "Phase locking and in-phase supermode selection in monolithic multicore fiber lasers," Opt. Lett. 31, 2577-2579 (2006).
    [CrossRef] [PubMed]
  10. C. J. Corcoran, and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A 7, L1-L7 (2005).
    [CrossRef]
  11. C. J. Corcoran and F. Durville, "Experimental demonatration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
    [CrossRef]
  12. D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, P. Even, and D. Pureur, "Efficient coherent combining of widely tunable fiber lasers," Opt. Express 11, 87-97 (2003).
    [CrossRef] [PubMed]
  13. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, "Coherent addition of fiber lasers by use of a fiber coupler," Opt. Express 10, 1167-1172 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-10-21-1167.
    [PubMed]
  14. H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, "200 W self-organized coherent fiber arrays," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonics Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CMDD4.
  15. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, "Self-organized coherence in fiber laser arrays," Opt. Lett. 30, 1339-1341 (2005).
    [CrossRef] [PubMed]
  16. S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
    [CrossRef]
  17. M. Fridman, V. Eckhouse, N. Davidson, and A. Friesem, "Efficient coherent addition of fiber lasers in free space," Opt. Lett. 32, 790-792 (2007).
    [CrossRef] [PubMed]
  18. B. Lei and Y. Feng, "Phase locking of an array of three fiber lasers by an all-fiber coupling loop," Opt. Express 15, 17114-17119 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-25-17114.
    [CrossRef] [PubMed]
  19. Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).
  20. Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
    [CrossRef]
  21. R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
    [CrossRef]
  22. J. Cao, Q. Lu, J. Hou, and X. Xu, "Dynamical model for self-organized fiber laser arrays" Opt. Express 17, 5402-5413 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-7-5402.
    [CrossRef] [PubMed]
  23. G. P. Agraval, Applications of nonlinear fiber optics (Elsevier Science, USA, 2001).

2009 (1)

2008 (2)

J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
[CrossRef]

2007 (4)

S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
[CrossRef]

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

M. Fridman, V. Eckhouse, N. Davidson, and A. Friesem, "Efficient coherent addition of fiber lasers in free space," Opt. Lett. 32, 790-792 (2007).
[CrossRef] [PubMed]

B. Lei and Y. Feng, "Phase locking of an array of three fiber lasers by an all-fiber coupling loop," Opt. Express 15, 17114-17119 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-25-17114.
[CrossRef] [PubMed]

2006 (4)

2005 (6)

T. Y. Fan, "Laser beam combining for high-power, high-radiance sources," IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005).
[CrossRef]

Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. 42, 26-29 (2005). (in Chinese)

C. J. Corcoran, and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A 7, L1-L7 (2005).
[CrossRef]

C. J. Corcoran and F. Durville, "Experimental demonatration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, "Self-organized coherence in fiber laser arrays," Opt. Lett. 30, 1339-1341 (2005).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

2000 (1)

1992 (1)

J. Morel, A. Woodtli, and R. Daendliker, "Coherent coupling of an array of Nd3+ doped single-mode fiber lasers using an intracavity phase grating," Proc. SPIE 1789,13-17 (1992).
[CrossRef]

Anderegg, J.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Aye, T. M.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Baker, J. T.

Barthelemy, A.

Benham, V.

Brosnan, S.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Bruesselbach, H.

Cao, J.

J. Cao, Q. Lu, J. Hou, and X. Xu, "Dynamical model for self-organized fiber laser arrays" Opt. Express 17, 5402-5413 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-7-5402.
[CrossRef] [PubMed]

J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)

Chen, S.

S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
[CrossRef]

L. Li, A. Schulzgen, S. Chen, and V. L. Temyanko, "Phase locking and in-phase supermode selection in monolithic multicore fiber lasers," Opt. Lett. 31, 2577-2579 (2006).
[CrossRef] [PubMed]

Chen, Z.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
[CrossRef]

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

Cheung, E.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Corcoran, C. J.

C. J. Corcoran and F. Durville, "Experimental demonatration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

C. J. Corcoran, and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A 7, L1-L7 (2005).
[CrossRef]

Culpepper, M. A.

Daendliker, R.

J. Morel, A. Woodtli, and R. Daendliker, "Coherent coupling of an array of Nd3+ doped single-mode fiber lasers using an intracavity phase grating," Proc. SPIE 1789,13-17 (1992).
[CrossRef]

Davidson, N.

Desfarges-Berthelemot, A.

DeShazer, L. G.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Durville, F.

C. J. Corcoran and F. Durville, "Experimental demonatration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

Eckhouse, V.

Epp, P.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Even, P.

Fan, D.

Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. 42, 26-29 (2005). (in Chinese)

Fan, T. Y.

T. Y. Fan, "Laser beam combining for high-power, high-radiance sources," IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005).
[CrossRef]

Feng, Y.

Fischer, D.

Fridman, M.

Friesem, A.

Glas, P.

Hammons, D.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Hou, J.

J. Cao, Q. Lu, J. Hou, and X. Xu, "Dynamical model for self-organized fiber laser arrays" Opt. Express 17, 5402-5413 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-7-5402.
[CrossRef] [PubMed]

J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
[CrossRef]

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

Jannson, T. P.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Jiang, Z.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
[CrossRef]

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

Jones, D. C.

Kermene, V.

Komine, H.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Kurt, R. M.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Lefort, L.

Lei, B.

Leitner, M.

Li, L.

Li, Y.

S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
[CrossRef]

Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. 42, 26-29 (2005). (in Chinese)

Liu, L.

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

Lu, C. A.

Lu, K.

S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
[CrossRef]

Lu, Q.

J. Cao, Q. Lu, J. Hou, and X. Xu, "Dynamical model for self-organized fiber laser arrays" Opt. Express 17, 5402-5413 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-7-5402.
[CrossRef] [PubMed]

J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)

Mangir, M. S.

Minden, M.

Morel, J.

J. Morel, A. Woodtli, and R. Daendliker, "Coherent coupling of an array of Nd3+ doped single-mode fiber lasers using an intracavity phase grating," Proc. SPIE 1789,13-17 (1992).
[CrossRef]

Napartovich, A. P.

Nelson, D. J.

Pasch, K. A.

C. J. Corcoran, and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A 7, L1-L7 (2005).
[CrossRef]

Pilkington, D.

Pradhan, R. D.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Pureur, D.

Rogers, J. L.

Sabourdy, D.

Sanchez, A. D.

Savant, G. D.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Schulzgen, A.

Shay, T. M.

Shirakawa,

Spring, J.

Temyanko, V. L.

Tun, N.

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

Vysotsky, D. V.

Ward, B.

Weber, M.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Wickham, M.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Woodtli, A.

J. Morel, A. Woodtli, and R. Daendliker, "Coherent coupling of an array of Nd3+ doped single-mode fiber lasers using an intracavity phase grating," Proc. SPIE 1789,13-17 (1992).
[CrossRef]

Wrage, M.

Xu, X.

J. Cao, Q. Lu, J. Hou, and X. Xu, "Dynamical model for self-organized fiber laser arrays" Opt. Express 17, 5402-5413 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-7-5402.
[CrossRef] [PubMed]

J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)

Zhou, P.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
[CrossRef]

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

Zhou, S.

S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
[CrossRef]

Acta Phys. Sin. (1)

Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, "Mutual injection locking of two individual fiber lasers," Acta Phys. Sin. 56, 7046-7050 (2007).

Appl. Phys. Lett. (1)

C. J. Corcoran and F. Durville, "Experimental demonatration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, "Mutual injection-locking and coherent combining of two individual fiber lasers," IEEE J. Quantum Electron. 44, 515-519 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

R. M. Kurt, R. D. Pradhan, N. Tun, T. M. Aye, G. D. Savant, T. P. Jannson, and L. G. DeShazer, "Mutual injection-locking: A new architecture for high-power solid-state laser arrays," IEEE J. Sel. Top. Quantum Electron. 11, 578-586 (2005).
[CrossRef]

T. Y. Fan, "Laser beam combining for high-power, high-radiance sources," IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005).
[CrossRef]

Infrared Laser Engin. (1)

J. Cao, X. Xu, J. Hou, and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Engin. 37, 456-460 (2008). (in Chinese)

J. Opt. A (2)

C. J. Corcoran, and K. A. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A 7, L1-L7 (2005).
[CrossRef]

S. Chen, Y. Li, K. Lu, and S. Zhou, "Efficient coherent combining of tunable erbium-doped fibre ring lasers," J. Opt. A 9, 642-648 (2007).
[CrossRef]

Laser Optoelectron. (1)

Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. 42, 26-29 (2005). (in Chinese)

Opt. Express (6)

Opt. Lett. (4)

Proc. SPIE (2)

J. Morel, A. Woodtli, and R. Daendliker, "Coherent coupling of an array of Nd3+ doped single-mode fiber lasers using an intracavity phase grating," Proc. SPIE 1789,13-17 (1992).
[CrossRef]

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," in Fiber Lasers III: Technology, Systems, and Applications.Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Other (2)

H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, "200 W self-organized coherent fiber arrays," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonics Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CMDD4.

G. P. Agraval, Applications of nonlinear fiber optics (Elsevier Science, USA, 2001).

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Figures (2)

Fig. 1.
Fig. 1.

Scheme of the array of two mutually-injected fiber lasers.

Fig. 2.
Fig. 2.

Fringe patterns of interference of two elementary lasers. Both of fringes are recorded with the same experimental arrangement. Each fringe corresponds to a phase-locked state. As is marked by the white dashed line, the position of maximum intensity of fringe (a) is identical with that of the minimum intensity of fringe (b), and vice versa.

Equations (31)

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

c d A m ( + ) ( x ) dx = g m ( x ) A m ( + ) ( x )
c d A m ( ) ( x ) dx = g m ( x ) A m ( ) ( x )
c d φ m ( + ) ( x ) dx = ( ω cm ω ) ( ϖ ω ) g m ( x ) γ
c d φ m ( ) ( x ) dx = ( ω cm ω ) ( ϖ ω ) g m ( x ) γ
γ / / [ g m ( x ) g 0 m ] g m ( x ) · { [ A m ( + ) ( x ) ] 2 + [ A m ( ) ( x ) ] 2 }
2 g m ( x ) A m ( + ) ( x ) A m ( ) ( x ) cos [ φ m ( + ) ( x ) φ m ( ) ( x ) + 2 k m x ] = 0
F cm = ( 1 ε m i ε m i ε m 1 ε m ) , ( m = 1,2 )
( A 1 ( ) ( L ) A 2 ( ) ( L ) ) = F co ( A 1 ( + ) ( L ) A 2 ( + ) ( L ) )
( A 1 ( + ) ( 0 ) A 2 ( + ) ( 0 ) ) = F fb ( A 1 ( ) ( 0 ) A 2 ( ) ( 0 ) )
F co = ( r 1 ( 1 ε 1 ) exp [ i 2 ω c ( l 1 ( c ) + l 1 ( r ) ) + ϕ R 1 ] ε 1 ε 2 exp [ i ω c ( l cc + l 1 ( c ) + l 2 ( c ) ) ] ε 1 ε 2 exp [ i ω c ( l cc + l 1 ( c ) + l 2 ( c ) ) ] r 2 ( 1 ε 2 ) exp [ i 2 ω c ( l 2 ( c ) + l 2 ( r ) ) + φ R 2 ] )
F fb = ( r ' 1 exp ( i 2 ω c l ' 1 ( r ) + ϕ ' R 1 ) 0 0 r ' 2 exp ( i 2 ω c l ' 2 ( r ) + ϕ ' R 2 ) )
A m ( ± ) ( 0 ) = A m ( ± ) ( 0 ) exp [ i φ m ( ± ) ( 0 ) ] , A m ( ± ) ( L ) = A m ( ± ) ( L ) exp [ i φ m ( ± ) ( L ) ] exp ( ± i k m L )
f m = A m ( ) ( L ) A m ( + ) ( L ) , ϑ m = φ m ( ) ( L ) φ m ( + ) ( L ) , ( m = 1,2 )
f 1 = { [ r 1 ( 1 ε 1 ) ] 2 + [ ε 1 ε 2 A 2 ( + ) ( L ) A 1 ( + ) ( L ) ] 2 + 2 r 1 ( 1 ε 1 ) ε 1 ε 2 A 2 ( + ) ( L ) A 1 ( + ) ( L ) cos θ 1 } 1 2
f 2 = { [ r 2 ( 1 ε 2 ) ] 2 + [ ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L ) ] 2 + 2 r 2 ( 1 ε 2 ) ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L ) cos θ 2 } 1 2
ϑ m = 2 ω c ( l m ( c ) + l m ( r ) ) + δ ϕ R m + Θ m , ( m = 1,2 )
θ 1 = ω c ( l cc + l 2 ( c ) l 1 ( c ) 2 l 1 ( r ) ) + ϕ 2 ( + ) ( L ) ϕ 1 ( + ) ( L ) + π δ ϕ R 1
θ 2 = ω c ( l cc + l 1 ( c ) l 2 ( c ) 2 l 2 ( r ) ) + ϕ 1 ( + ) ( L ) ϕ 2 ( + ) ( L ) + π δ ϕ R 2
Θ 1 = arctan { ε 1 ε 2 A 2 ( + ) ( L , t ) A 1 ( + ) ( L , t ) sin ( θ 1 ) r 1 ( 1 ε 1 ) + ε 1 ε 2 A 2 ( + ) ( L , t ) A 1 ( + ) ( L , t ) cos ( θ 1 ) }
Θ 2 = arctan { ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L ) sin ( θ 2 ) r 2 ( 1 ε 2 ) + ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L ) cos ( θ 2 ) }
θ 1 + θ 2 = 2 , θ 1 = 2 n 1 π , ( n , n 1 = 0 , ± 1 , ± 2 , )
ω c ( 2 l cc 2 l 1 ( r ) 2 l 2 ( r ) ) δ ϕ R 1 δ ϕ R 2 = 2 n π
ω c ( l cc + l 2 ( c ) l 1 ( c ) 2 l 1 ( r ) ) + ϕ 2 ( + ) ( L ) ϕ 1 ( + ) ( L ) + π δ ϕ R 1 = 2 n 1 π
Δ θ 21 = θ 2 θ 1 = q 1 π 1 2 ( δ ϕ R 2 δ ϕ R 1 ) , ( q 1 = 0 , ± 1 , ± 2 , )
q 1 = { 0 , ± 2 , ± 4 , ± 6 , ± 8 , ( State I ) ± 1 , ± 3 , ± 5 , ± 7 , ( State II )
R t 1 = 2 r 1 ( 1 ε 1 ) ε 1 ε 2 A 2 ( + ) ( L ) A 1 ( + ) ( L ) [ r 1 ( 1 ε 1 ) ] 2 + [ ε 1 ε 2 A 2 ( + ) ( L ) A 1 ( + ) ( L ) ] 2 , R t 2 = 2 r 2 ( 1 ε 2 ) ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L ) [ r 2 ( 1 ε 2 ) ] 2 + [ ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L ) ] 2
r 1 ( 1 ε 1 ) = ε 1 ε 2 A 2 ( + ) ( L ) A 1 ( + ) ( L ) , r 2 ( 1 ε 2 ) = ε 1 ε 2 A 1 ( + ) ( L ) A 2 ( + ) ( L )
ε = r 1 + r
Δ Φ 1 = ω c L ¯ 1 + δ ϕ R 1 + Θ 1 + ( ϖ ω ) γ ln [ f 1 r 1 ' ] + δ ϕ ' R 1 = 2 n 3 π
Δ Φ 2 = ω c L ¯ 2 + δ ϕ R 2 + Θ 2 + ( ϖ ω ) γ ln [ f 2 r 2 ' ] + δ ϕ ' R 2 = 2 n 4 π
Δ Φ 2 Δ Φ 1 = 2 n 5 π , ( n 3 , n 4 , n 5 = 0 , ± 1 , ± 2 , ± 3 , )

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