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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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).
  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 and Laser Engineering 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]
  5. 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).
    [PubMed]
  6. Shirakawa, K. Matsuo, and K. Ueda, “Fiber-laser coherent array for power scaling of single-mode fiber laser,” Proc. SPIE 5662, 482–487 (2004).
  7. 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).
  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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  10. 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]
  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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    [PubMed]
  14. Bing Lei and Ying Feng, “Phase locking of an array of three fiber lasers by an all-fiber coupling loop” Opt. Express 15, 17114–17119 (2007).
    [PubMed]
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  16. 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).
  17. J. L. Rogers, S. Pelěs, and K. Wiesenfeld, “Model for High-Gain Fiber Laser Arrays,” IEEE J. Quantum Electron.  41, 767–773 (2005).
  18. S. Pelěs, 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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    [PubMed]
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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).
    [PubMed]
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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. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
  26. C. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, New York, 1991).
  27. G. P. Agraval, Applications of nonlinear fiber optics (Elsevier Science, USA, 2001).

2008 (3)

J. Cao, X. Xu, J. Hou, and Q. Lu, “Coheret combining technology of fiber laser,” Infrared and Laser Engineering 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).

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

2007 (3)

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).

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

Bing Lei and Ying Feng, “Phase locking of an array of three fiber lasers by an all-fiber coupling loop” Opt. Express 15, 17114–17119 (2007).
[PubMed]

2006 (2)

S. Pelěs, 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 (7)

J. L. Rogers, S. Pelěs, and K. Wiesenfeld, “Model for High-Gain Fiber Laser Arrays,” IEEE J. Quantum Electron.  41, 767–773 (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)

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).

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.

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)

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).

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

2003 (1)

2002 (1)

1999 (2)

J. Terry, K. Thornburg, 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).

E. M. Pessina, F. Prati, J. Redondo, E. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).

1997 (1)

K. Thornberg, 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]

Agraval, G. P.

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

Ashwin, P.

J. Terry, K. Thornburg, 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, 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.

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, X. Xu, J. Hou, and Q. Lu, “Coheret combining technology of fiber laser,” Infrared and Laser Engineering 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).

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.

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

de Valcärcel, G. J.

E. M. Pessina, F. Prati, J. Redondo, E. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).

Desfarges-Berthelemot, A.

DeShazer, D.

J. Terry, K. Thornburg, 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.

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

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).

Feng, Ying

Fridman, M.

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

Friesem, A.

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

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, X. Xu, J. Hou, and Q. Lu, “Coheret combining technology of fiber laser,” Infrared and Laser Engineering 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).

Jones, D. C.

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.

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.

Lei, Bing

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).

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

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

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, 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, 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 and Laser Engineering 37, 456–460 (2008). (in Chinese)

Mangir, M. S.

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.

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).

Matsuo, K.

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).

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

Minden, M.

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.

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]

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, M. Moller, and R. Roy, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).

Peles, S.

S. Pelěs, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E 73, 026212 (2006).

J. L. Rogers, S. Pelěs, and K. Wiesenfeld, “Model for High-Gain Fiber Laser Arrays,” IEEE J. Quantum Electron.  41, 767–773 (2005).

Pessina, E. M.

E. M. Pessina, F. Prati, J. Redondo, E. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).

Prati, F.

E. M. Pessina, F. Prati, J. Redondo, E. Roldän, and G. J. de Valcärcel, “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. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).

Rogers, J. L.

S. Pelěs, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E 73, 026212 (2006).

J. L. Rogers, S. Pelěs, and K. Wiesenfeld, “Model for High-Gain Fiber Laser Arrays,” IEEE J. Quantum Electron.  41, 767–773 (2005).

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.

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).

Roldän, E.

E. M. Pessina, F. Prati, J. Redondo, E. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).

Roy, R.

J. Terry, K. Thornburg, 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, 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.

Saitou, T.

Sekiguchi, T.

Shirakawa,

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).

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

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).
[PubMed]

Siegman, A. E.

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

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. Thornburg, 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, M. Moller, and R. Roy, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).

Thornburg, K.

J. Terry, K. Thornburg, 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).

Ueda, K.

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).

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

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).
[PubMed]

VanWiggeren, G.

J. Terry, K. Thornburg, 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).

Vilaseca, R.

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

Weiss, C.

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

Wiesenfeld, K.

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

S. Pelěs, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E 73, 026212 (2006).

J. L. Rogers, S. Pelěs, and K. Wiesenfeld, “Model for High-Gain Fiber Laser Arrays,” IEEE J. Quantum Electron.  41, 767–773 (2005).

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

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

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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).

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. Thornburg, 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. Pelěs, 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 and Laser Engineering 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)

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

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]

Phys. Rev. A (3)

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).
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E. M. Pessina, F. Prati, J. Redondo, E. Roldän, and G. J. de Valcärcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).

Phys. Rev. E (4)

J. Terry, K. Thornburg, 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, M. Moller, and R. Roy, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).

S. Pelěs, 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)

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

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).

Technical Digest (CD) (1)

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.

Other (3)

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

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

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

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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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