Abstract

A dynamics model of self-organized fiber laser arrays is presented in this paper. The model does not only break the limitation of the standard slowly varying wave approximation, but also be built on the basis of Maxwell-Bloch equations which make this model more suitable to study the dynamics (especially phase dynamics) of fiber laser arrays. In this paper, this model is applied to analyze fiber laser array of interferometric configuration. The results agree well with the reported experimental results. It is also revealed that the coupling strength of 2-fiber laser array of interferometric configuration have a negligible effect on the phase-locked state of the array.

© 2009 Optical Society of America

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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).
  2. Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. Prog. 42, 26-29 (2005). (in Chinese)
  3. J. Cao, X. Xu, J. Hou and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Eng. 37, 456-460 (2008). (in Chinese)
  4. 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).
    [PubMed]
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    [PubMed]
  6. A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber-laser coherent array for power scaling of single-mode fiber laser," Proc. SPIE 5662, 482-487 (2004).
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  8. M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).
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    [PubMed]
  11. S. Chen, Y. Li, and K. Lu, "Branch arm filtered coherent combining of tunable fiber lasers," Opt. Express 13, 7878-7883 (2005).
    [PubMed]
  12. 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).
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  17. J. L. Rogers, S. Pele¡s, and K. Wiesenfeld, "Model for High-Gain Fiber Laser Arrays," IEEE J. Quantum Electron. 41, 767-773 (2005).
  18. S. Pele¡s, J. L. Rogers, and K. Wiesenfeld, "Robust synchronization in fiber laser arrays," Phys. Rev. E 73, 026212 (2006).
  19. D. Tsygankov and K. Wiesenfeld, "Weak-link synchronization," Phys. Rev. E 73, 026222 (2006).
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  22. Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
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  24. J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).
  25. E. M. Pessina, F. Prati, J. Redondo, E. Rolda’n, and G. J. de Valca’rcel, "Multimode instability in ring fiber lasers," Phys. Rev. A 60, 2517- 2528 (1999).
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2008 (3)

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

J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).

J. Cao, J. Hou, Q. Lu, and X. Xu, "Numerical research on self-organized coherent fiber laser arrays with circulating field theory," J. Opt. Soc. Am. B 25, 1187-1192 (2008).

2007 (3)

2006 (2)

S. Pele¡s, J. L. Rogers, and K. Wiesenfeld, "Robust synchronization in fiber laser arrays," Phys. Rev. E 73, 026212 (2006).

D. Tsygankov and K. Wiesenfeld, "Weak-link synchronization," Phys. Rev. E 73, 026222 (2006).

2005 (6)

A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control," Proc. SPIE 5709, 165-174 (2005).

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

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

J. L. Rogers, S. Pele¡s, and K. Wiesenfeld, "Model for High-Gain Fiber Laser Arrays," IEEE J. Quantum Electron. 41, 767-773 (2005).

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).
[PubMed]

S. Chen, Y. Li, and K. Lu, "Branch arm filtered coherent combining of tunable fiber lasers," Opt. Express 13, 7878-7883 (2005).
[PubMed]

2004 (2)

A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber-laser coherent array for power scaling of single-mode fiber laser," Proc. SPIE 5662, 482-487 (2004).

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

2003 (1)

2002 (1)

1999 (2)

E. M. Pessina, F. Prati, J. Redondo, E. Rolda’n, and G. J. de Valca’rcel, "Multimode instability in ring fiber lasers," Phys. Rev. A 60, 2517- 2528 (1999).

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

1997 (1)

K. Thornberg Jr., M. Moller, and R. Roy, "Chaos and coherence in coupled lasers," Phys. Rev. E 55, 3865-3869 (1997).

1995 (1)

Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
[PubMed]

1993 (1)

L. Fabiny, P. Colet, and R. Roy, "Coherence and phase dynamics of spatially coupled solid-state lasers," Phys. Rev. A 47, 4287- 4296 (1993).
[PubMed]

Ashwin, P.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

Barthelemy, A.

Braiman, Y.

Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
[PubMed]

Bruesselbach, H.

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).
[PubMed]

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Cao, J.

J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).

J. Cao, J. Hou, Q. Lu, and X. Xu, "Numerical research on self-organized coherent fiber laser arrays with circulating field theory," J. Opt. Soc. Am. B 25, 1187-1192 (2008).

J. Cao, X. Xu, J. Hou and Q. Lu, "Coheret combining technology of fiber laser," Infrared Laser Eng. 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).

S. Chen, Y. Li, and K. Lu, "Branch arm filtered coherent combining of tunable fiber lasers," Opt. Express 13, 7878-7883 (2005).
[PubMed]

Colet, P.

L. Fabiny, P. Colet, and R. Roy, "Coherence and phase dynamics of spatially coupled solid-state lasers," Phys. Rev. A 47, 4287- 4296 (1993).
[PubMed]

Davidson, N.

Desfarges-Berthelemot, A.

DeShazer, D.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

Dunnings, G. J.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Eckhouse, V.

Even, P.

Fabiny, L.

L. Fabiny, P. Colet, and R. Roy, "Coherence and phase dynamics of spatially coupled solid-state lasers," Phys. Rev. A 47, 4287- 4296 (1993).
[PubMed]

Fan, D.

Y. Li and D. Fan, "Beam combining of fiber laser," Laser Optoelectron. Prog. 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).

Fridman, M.

Friesem, A.

Hammon, D. L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Hou, J.

J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).

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

J. Cao, J. Hou, Q. Lu, and X. Xu, "Numerical research on self-organized coherent fiber laser arrays with circulating field theory," J. Opt. Soc. Am. B 25, 1187-1192 (2008).

Jones, D. C.

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).
[PubMed]

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Kennedy, T.

Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
[PubMed]

Kermene, V.

Khibnik, A.

Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
[PubMed]

Lefort, 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).

S. Chen, Y. Li, and K. Lu, "Branch arm filtered coherent combining of tunable fiber lasers," Opt. Express 13, 7878-7883 (2005).
[PubMed]

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

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).

S. Chen, Y. Li, and K. Lu, "Branch arm filtered coherent combining of tunable fiber lasers," Opt. Express 13, 7878-7883 (2005).
[PubMed]

Lu, Q.

J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).

J. Cao, J. Hou, Q. Lu, and X. Xu, "Numerical research on self-organized coherent fiber laser arrays with circulating field theory," J. Opt. Soc. Am. B 25, 1187-1192 (2008).

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

Mangir, M. S.

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).
[PubMed]

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Minden, M.

Minden, M. L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Moller, M.

K. Thornberg Jr., M. Moller, and R. Roy, "Chaos and coherence in coupled lasers," Phys. Rev. E 55, 3865-3869 (1997).

Pessina, E. M.

E. M. Pessina, F. Prati, J. Redondo, E. Rolda’n, and G. J. de Valca’rcel, "Multimode instability in ring fiber lasers," Phys. Rev. A 60, 2517- 2528 (1999).

Prati, F.

E. M. Pessina, F. Prati, J. Redondo, E. Rolda’n, and G. J. de Valca’rcel, "Multimode instability in ring fiber lasers," Phys. Rev. A 60, 2517- 2528 (1999).

Pureur, D.

Redondo, J.

E. M. Pessina, F. Prati, J. Redondo, E. Rolda’n, and G. J. de Valca’rcel, "Multimode instability in ring fiber lasers," Phys. Rev. A 60, 2517- 2528 (1999).

Rogers, J. L.

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).
[PubMed]

J. L. Rogers, S. Pele¡s, and K. Wiesenfeld, "Model for High-Gain Fiber Laser Arrays," IEEE J. Quantum Electron. 41, 767-773 (2005).

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Roy, R.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

K. Thornberg Jr., M. Moller, and R. Roy, "Chaos and coherence in coupled lasers," Phys. Rev. E 55, 3865-3869 (1997).

L. Fabiny, P. Colet, and R. Roy, "Coherence and phase dynamics of spatially coupled solid-state lasers," Phys. Rev. A 47, 4287- 4296 (1993).
[PubMed]

Sabourdy, D.

Shirakawa,

Shirakawa, A.

A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control," Proc. SPIE 5709, 165-174 (2005).

A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber-laser coherent array for power scaling of single-mode fiber laser," Proc. SPIE 5662, 482-487 (2004).

Solis, A. J.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Terry, J.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

Thornberg, K.

K. Thornberg Jr., M. Moller, and R. Roy, "Chaos and coherence in coupled lasers," Phys. Rev. E 55, 3865-3869 (1997).

Thornburg, K.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

Tsygankov, D.

D. Tsygankov and K. Wiesenfeld, "Weak-link synchronization," Phys. Rev. E 73, 026222 (2006).

VanWiggeren, G.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

Vaughan, L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Wiesenfeld, K.

D. Tsygankov and K. Wiesenfeld, "Weak-link synchronization," Phys. Rev. E 73, 026222 (2006).

Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
[PubMed]

Xu, X.

J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).

J. Cao, J. Hou, Q. Lu, and X. Xu, "Numerical research on self-organized coherent fiber laser arrays with circulating field theory," J. Opt. Soc. Am. B 25, 1187-1192 (2008).

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

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).

Zhu, S.

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

Appl. Phys. B (1)

J. Cao, Q. Lu, X. Xu, and J. Hou, "Chaos in coupled lasers with low-frequency modulation," Appl. Phys. B 92, 525-528 (2008).

IEEE J. Quantum Electron. (1)

J. L. Rogers, S. Pele¡s, and K. Wiesenfeld, "Model for High-Gain Fiber Laser Arrays," IEEE J. Quantum Electron. 41, 767-773 (2005).

IEEE J. Sel. Top. Quantum Electron. (1)

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

Infrared and Laser Engineering (1)

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

J. Opt. A (1)

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).

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

Laser Optoelectron. Prog. (1)

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

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (3)

E. M. Pessina, F. Prati, J. Redondo, E. Rolda’n, and G. J. de Valca’rcel, "Multimode instability in ring fiber lasers," Phys. Rev. A 60, 2517- 2528 (1999).

L. Fabiny, P. Colet, and R. Roy, "Coherence and phase dynamics of spatially coupled solid-state lasers," Phys. Rev. A 47, 4287- 4296 (1993).
[PubMed]

Y. Braiman, T. Kennedy, K. Wiesenfeld, and A. Khibnik, "Entrainment of solid-state laser arrays," Phys. Rev. A 52, 1500- 1506 (1995).
[PubMed]

Phys. Rev. E (4)

J. Terry, K. ThornburgJr., D. DeShazer, G. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, "Synchronization of chaos in an array of three lasers," Phys. Rev. E 59, 4036- 4043 (1999).

K. Thornberg Jr., M. Moller, and R. Roy, "Chaos and coherence in coupled lasers," Phys. Rev. E 55, 3865-3869 (1997).

S. Pele¡s, J. L. Rogers, and K. Wiesenfeld, "Robust synchronization in fiber laser arrays," Phys. Rev. E 73, 026212 (2006).

D. Tsygankov and K. Wiesenfeld, "Weak-link synchronization," Phys. Rev. E 73, 026222 (2006).

Proc. SPIE (3)

A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber-laser coherent array for power scaling of single-mode fiber laser," Proc. SPIE 5662, 482-487 (2004).

A. Shirakawa, K. Matsuo, and K. Ueda, "Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control," Proc. SPIE 5709, 165-174 (2005).

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunnings, D. L. Hammon, A. J. Solis, and L. Vaughan, "Self-organized coherence in fiber laser arrays," Proc. SPIE 5335, 89-97 (2004).

Other (4)

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.

C. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, New York, 1991).

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

A. E. Siegman, "Resonant modes of linearly coupled multiple fiber laser structures," http://www.stanford.edu/~siegman/coupled_fiber_modes.pdf.

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Figures (2)

Fig. 1.
Fig. 1.

Scheme of fiber laser arrays

Fig. 2.
Fig. 2.

Scheme of 2-laser interferometric array

Equations (68)

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E ˙ m ( x , t ) i c 2 2 ω 2 E m ( x , t ) x 2 i ω 2 E m ( x , t ) = 2 ε 0 P m ( x , t )
P ˙ m ( x , t ) = [ i ( ϖ ω ) γ ] P m ( x , t ) + 1 E m ( x , t ) θ 12 2 D m ( x , t )
D ˙ m ( x , t ) = γ ( D m D 0 m ) 2 [ E m ( x , t ) P m * ( x , t ) E m * ( x , t ) P m ( x , t ) ]
E m ( x , t ) = E m ( + ) ( x , t ) exp ( i k m x ) + E m ( ) ( x , t ) exp ( i k m x )
P m ( x , t ) = P m ( + ) ( x , t ) exp ( i k m x ) + P m ( ) ( x , t ) exp ( i k m x )
c E m ( + ) ( x , t ) x + E ˙ m ( + ) ( x , t ) = i ( ω cm ω ) E m ( + ) ( x , t ) + 2 ε 0 P m ( + ) ( x , t )
c E m ( ) ( x , t ) x + E ˙ m ( ) ( x , t ) = i ( ω cm ω ) E m ( ) ( x , t ) + 2 ε 0 P m ( ) ( x , t )
P ˙ m ( + ) ( x , t ) = [ i ( ϖ ω ) γ ] P m ( + ) ( x , t ) + 1 E m ( + ) ( x , t ) θ 12 2 D m ( x , t )
P ˙ m ( ) ( x , t ) = [ i ( ϖ ω ) γ ] P m ( ) ( x , t ) + 1 E m ( ) ( x , t ) θ 12 2 D m ( x , t )
D ˙ m ( x , t ) = γ ( D m D 0 m ) 2 [ E m ( x , y ) P m * ( x , t ) E m * ( x , t ) P m ( x , t ) ]
c E m ( + ) ( x , t ) x + E m ( + ) ( x , t ) = i ( ω cm ω ) E m ( + ) ( x , t ) + g m ( x , t ) E m ( + ) ( x , t ) i ( ϖ ω ) g ( x , t ) γ E m ( + ) ( x , t )
c E m ( ) ( x , t ) x + E ˙ m ( ) ( x , t ) = i ( ω cm ω ) E m ( ) ( x , t ) + g m ( x , t ) E m ( ) ( x , t ) i ( ϖ ω ) g ( x , t ) γ E m ( ) ( x , t )
g ˙ m ( x , t ) = γ [ g m ( x , t ) g 0 m ( x , t ) ] 4 c σ ε 0 ωħ g m ( x , t ) ×
{ E m ( + ) ( x , t ) 2 + E m ( ) ( x , t ) 2 + E m ( ) ( x , t ) [ E m ( + ) ( x , t ) ] * exp ( 2 ikx ) + E m ( + ) ( x , t ) [ E m ( ) ( x , t ) ] * exp ( 2 ikx ) }
E m ( + ) ( x , t ) = E m ( + ) ( x , t ) exp [ i ϕ m ( + ) ( x , t ) ]
E m ( ) ( x , t ) = E m ( ) ( x , t ) exp [ i ϕ m ( ) ( x , t ) ]
A m ( + ) ( x , t ) = 4 ε 0 ωħ E m ( + ) ( x , t ) , A m ( ) ( x , t ) = 4 ε 0 ωħ E m ( ) ( x , t )
c A m ( + ) ( x , t ) x + A m ( + ) ( x , t ) t = g m ( x , t ) A m ( + ) ( x , t )
c A m ( ) ( x , t ) x + A m ( ) ( x , t ) t = g m ( x , t ) A m ( ) ( x , t )
c ϕ m ( + ) ( x , t ) x + ϕ m ( + ) ( x , t ) t = ( ω cm ω ) ( ϖ ω ) g m ( x , t ) γ
c ϕ m ( ) ( x , t ) x + ϕ m ( ) ( x , t ) t = ( ω cm ω ) ( ϖ ω ) g m ( x , t ) γ
g ˙ m ( x , t ) = γ [ g m ( x , t ) g 0 m ( x , t ) ] g m ( x , t ) ×
{ [ A m ( + ) ( x , t ) ] 2 + [ A m ( ) ( x , t ) ] 2 + 2 A m ( + ) ( x , t ) A m ( ) ( x , t ) cos [ ϕ m ( + ) ( x , t ) ϕ m ( ) ( x , t ) + 2 kx ] }
A m ( + ) ( L , t ) = A m ( + ) ( L , t ) exp [ i ϕ m ( + ) ( L , t ) ] exp ( i k m L )
A m ( + ) ( L c , t ) = A m ( + ) ( L , t ) exp ( i ω c l m ( c ) )
F c = { C m , n } m , n = 1 m , n = N
A ' m ( + ) ( L c , t ) = n = 1 N C m , n A n ( + ) ( L c , t )
A m ( ) ( L c , t ) = r m e φ Rm exp ( i 2 ω c l m ( r ) ) A ' m ( + ) ( L c , t )
A ' m ( ) ( L c , t ) = n = 1 N C m , n A n ( ) ( L c , t )
A m ( ) ( L , t ) = A ' m ( ) ( L c , t ) exp ( i ω c l m ( c ) )
A m ( ) ( L , t ) = A m ( ) ( L , t ) exp [ i ϕ m ( ) ( L , t ) ] exp ( i k m L )
A 1 ( ) ( L , t ) A 2 ( ) ( L , t ) A m ( ) ( L , t ) A N ( ) ( L , t ) = F A 1 ( + ) ( L , t ) A 2 ( + ) ( L , t ) A m ( + ) ( L , t ) A N ( + ) ( L , t )
F = F P F c F R F c F p
( F p ) mm = exp ( i ω c l m ( c ) ) , ( F p ) jm = 0 , ( j , m = 1,2 , N ; j m )
( F R ) mm = r m e φ Rm exp ( i 2 ω c l m ( r ) ) , ( F R ) jm = 0 , ( j , m = 1,2 , , N ; j m )
A 1 ( + ) ( 0 , t ) A 2 ( + ) ( 0 , t ) A m ( + ) ( 0 , t ) A N ( + ) ( 0 , t ) = F A 1 ( ) ( 0 , t ) A 2 ( ) ( 0 , t ) A m ( ) ( 0 , t ) A N ( ) ( 0 , t )
( F ' ) mm = r ' m e φ ' Rm exp ( i 2 ω c l ' m ( r ) ) , ( F ' ) jm = 0 , ( j , m = 1,2 , , N ; j m )
A m ( + ) ( 0 , t ) = A m ( + ) ( 0 , t ) exp [ i ϕ m ( + ) ( 0 , t ) ] , A m ( ) ( 0 , t ) = A m ( ) ( 0 , t ) exp [ i ϕ m ( ) ( 0 , t ) ]
F R = exp ( i 2 ω c l 1 ( r ) ) re φ R 0 0 0
F c = 1 ε i ε i ε 1 ε
F = r exp [ i ( δ ϕ R + 2 ω c l 1 ( r ) ) ] ( 1 ε ) exp ( i 2 ω c l 1 ( c ) ) ε ( 1 ε ) exp [ i ω c ( l 1 ( c ) + l 2 ( c ) ) + i π 2 ] ε ( 1 ε ) exp [ i ω c ( l 1 ( c ) + l 2 ( c ) ) + i π 2 ] ε exp [ i ( 2 ω c l 2 ( c ) π ) ]
f 1 = A 1 ( ) ( L ) A 1 ( + ) ( L ) , f 2 = A 2 ( ) ( L ) A 2 ( + ) ( L )
ϑ 1 = ϕ 1 ( ) ( L ) ϕ 1 ( + ) ( L ) , ϑ 2 = ϕ 2 ( ) ( L ) ϕ 2 ( + ) ( L ) .
c Δ ϕ 21 st ( + ) ( x ) x = ( ω c 2 ω c 1 ) ( ϖ ω ) γ [ g 2 st ( x ) g 1 st ( x ) ]
c Δ ϕ 21 st ( + ) ( x , t ) x = ( ω c 2 ω c 1 ) ( ϖ ω ) γ [ g 2 st ( x ) g 1 st ( x ) ]
Δ ϕ 21 st ( + ) ( L ) Δ ϕ 21 st ( + ) ( 0 ) = ( ω c 2 ω c 1 ) c L ( ϖ ω ) γ { ln [ A 2 st ( + ) ( L ) A 2 st ( + ) ( 0 ) ] ln [ A 1 st ( + ) ( L ) A 1 st ( + ) ( 0 ) ] }
Δ ϕ 21 st ( ) ( 0 ) Δ ϕ 21 st ( ) ( L ) = ( ω c 2 ω c 1 ) c L + ( ϖ ω ) γ { ln [ A 2 st ( ) ( L ) A 2 st ( ) ( 0 ) ] ln [ A 1 st ( ) ( L ) A 1 st ( ) ( 0 ) ] }
Δ ϕ 21 st ( ) ( L ) = ω c ( l 2 ( c ) l 1 ( c ) ) + π 2 ω c 2 l 2 ( c ) c ω c 1 l 1 ( c ) c
Δ ϕ 21 st ( + ) ( 0 ) = Δ ϕ 21 st ( ) ( 0 ) = 2 ω c ( l ' 2 ( r ) l ' 1 ( r ) ) 2 ω c 2 l ' 2 ( r ) c + 2 ω c 1 l ' 1 ( r ) c + ( δ φ ' R 2 δ φ ' R 1 )
Δ ϕ 21 st ( + ) ( L ) = ω c [ ( l 2 ( c ) + 2 l ' 2 ( r ) ) ( l 1 ( c ) + 2 l ' 1 ( r ) ) ]
ω c 2 L ' ¯ 2 c + ω c 1 L ' ¯ 1 c + π 2 + ( δ φ ' R 2 δ φ ' R 1 ) + ( ϖ ω ) γ ln ( f 2 r ' 2 f 1 r ' 1 )
A st ( + ) ( L c ) = 1 ε A 1 st ( + ) ( L ) + ε A 2 st ( + ) ( L ) e Δ φ 21 ( + )
Δ φ 21 ( + ) = 2 ω c [ ( l 2 ( c ) + l ' 2 ( r ) ) ( l 1 ( c ) + l ' 1 ( r ) ) ] + ( ϖ ω ) γ ln ( r ' 2 r ' 1 ε 1 ε ) + π + ( δ φ ' R 2 δ φ ' R 1 )
ε = 1 2 , r ' 1 = r ' 2 = 1 , δ φ R = δ φ ' R 1 = δ φ ' R 2 = π
Δ φ 21 ( + ) = 2 ω c [ ( l 2 ( c ) + l ' 2 ( r ) ) ( l 1 ( c ) + l ' 1 ( r ) ) ] + π
Δ φ 21 ( + ) = 2 , ( q = 0 , ± 1 , ± 2 , )
Δ ω = πc ( l 2 ( c ) + l ' 2 ( r ) ) ( l 1 ( c ) + l ' 1 ( r ) )
A 1 ( ) ( L , t + τ 1 ) A 2 ( ) ( L , t + τ 2 ) A m ( ) ( L , t + τ m ) A N ( ) ( L , t + τ N ) = F A 1 ( + ) ( L , t ) A 2 ( + ) ( L , t ) A m ( + ) ( L , t ) A N ( + ) ( L , t )
A 1 ( ) ( L , t ) = B 1 1 ε A 1 ( + ) ( L , t ) + i ε A 2 ( + ) ( L , t ) exp ( )
A 2 ( ) ( L , t ) = iB 2 1 ε A 1 ( + ) ( L , t ) + i ε A 2 ( + ) ( L , t ) exp ( )
B 1 = r 1 ε exp { i [ δ ϕ R + 2 ω c ( l 1 ( r ) + l 1 ( c ) ) + ω c 1 c L + ϕ 1 ( + ) ( L , t ) ] }
B 2 = r ε exp { i [ δ ϕ R + ω c ( l 1 ( c ) + l 2 ( c ) + 2 l 1 ( r ) ) + ω c 1 c L + ϕ 1 ( + ) ( L , t ) ] }
θ = ω c ( l 2 ( c ) l 1 ( c ) ) + ϕ 2 ( + ) ( L , t ) ϕ 1 ( + ) ( L , t ) + ( ω c 2 ω c 1 ) c L
f 1 = r ( 1 ε ) { 1 2 ε 1 ε [ A 2 ( + ) ( L , t ) A 1 ( + ) ( L , t ) ] sin θ + ( ε 1 ε ) [ A 2 ( + ) ( L , t ) A 1 ( + ) ( L , t ) ] 2 } 1 2
ϑ 1 = δ ϕ R + 2 ω c ( l 1 ( r ) + l 1 ( c ) ) + 2 ω c 1 c L + Θ
f 2 = r ε ( 1 ε ) { 1 2 ε 1 ε [ A 2 ( + ) ( L , t ) A 1 ( + ) ( L , t ) ] sin θ + ( ε 1 ε ) [ A 2 ( + ) ( L , t ) A 1 ( + ) ( L , t ) ] 2 } 1 2
ϑ 2 = δ ϕ R + ω c ( l 1 ( c ) + l 2 ( c ) + 2 l 1 ( r ) ) + ω c 1 c L + ω c 2 c L + π 2 + Θ ϕ 1 ( + ) ( L , t ) ϕ 2 ( + ) ( L , t )
f 2 f 1 = ε 1 ε

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