## Self-organization of arrays of two mutually-injected fiber lasers: theoretical investigation

Optics Express, Vol. 17, Issue 9, pp. 7694-7701 (2009)

http://dx.doi.org/10.1364/OE.17.007694

Acrobat PDF (212 KB)

### 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

## 1. Introduction

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

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]

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]

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]

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]

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]

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]

## 2. Description of the array of mutually-injected fiber lasers

*A*

_{m}^{(±)}(

*x*) and

*φ*

_{m}^{(±)}(

*x*) are the amplitude (arbitrary unit) and phase (unit: rad) of the complex slow-varying envelope of laser fields. The superscripts “+” and “ − ” correspond, respectively, to right-going and left-going fields propagating in the array. The subscript

*m*=1, 2 represent two fiber lasers.

*g*(

_{m}*x*) is the gain coefficient (unit: s

^{-1}), and

*g*

_{0m}is the pump coefficient (unit: s

^{-1}).

*ω*is the laser frequency (unit: rad/s),

*ϖ*is the atomic frequency (unit:rad/s).

*ω*−

_{cm}*ck*is the resonance frequency (unit: rad/s) of the

_{m}*m*th cavity,

*C*is the velocity of light in vacuum (unit: m/s),

*k*is the wave vector (unit: rad/m).

_{m}*γ*

_{⊥}and

*γ*are transverse and longitudinal relaxation rates (unit: s

_{∥}^{-1}), respectively.

*ε*is the percentage of the input power coupled crossly into the other port [23]. Then, two boundary conditions corresponding to coupling-output and feedback sections are

_{m}*r*(or

_{m}*r*'

*) is the reflectivity coefficient of the output face (or FBG) of the*

_{m}*m*th laser cavity;

*δϕ*(

_{Rm}*δϕ'*) are the phase differences caused by the reflecting of the output face (FBG).

_{Rm}*l*

_{m}^{(c)}is the optical path length (OPL) between

*x*=

*L*and the coupler of

*m*th cavity;

*l*

_{m}^{(r)}is the OPL between the coupler and the output face;

*l*'

_{m}^{(r)}is the OPL between

*x*= 0 and FBG;

*l*is the OPL between two couplers (i.e., C-1 and C-2). Note that the fields

_{cc}**A**

_{m}^{(±)}(0) and

**A**

_{m}^{(±)}(

*L*) should be expressed as

## 3. Results and discussions

*f*and the phase difference

_{m}*ϑ*caused by the coupler-output section. Here,

_{m}*f*and

_{m}*ϑ*are defined as

_{m}*ϕ*

_{m}^{(±)}(

*L*) =

*φ*

_{m}^{(±)}(

*L*) ±

*km*.

_{L}*f*varies with the laser frequency

_{m}*ω*, because

*θ*is the function of

_{m}*ω*. It means that the longitudinal modes own different losses in the coupler-output section. According to the mode-competition theory, the longitudinal mode(s) corresponding to the minimum loss (i.e., the maximum equivalent reflectivity) will be dominant in the mode-competition and oscillate in the array. Therefore, the longitudinal mode(s) should satisfy the maximum conditions of

*f*

_{1}and

*f*

_{2}, i.e.,

*θ*

_{21}is just the phase difference of two elementary lasers at the output faces. Meanwhile, the phase difference is a constant which does not vary with the OPL of fibers in the array. It means that two elementary lasers are phase-locked at the output faces.

*f*on

_{m}*ω*which is determined by the cosine term of

*θ*in the expression of fm (see Eqs. (13) and (14) and Eqs. (16) and (17)). Therefore, the weight of the cosine term will determine the longitudinal-mode discrimination of the array, and ultimately, influence the output state of the array. If the weight of the cosine term is too small, the longitudinal-mode discrimination will be too weak to realize effective self-organization. As a result, the coherence of elementary beams will decrease and the visibility of the interference fringe will be lowered.

_{m}*Rt*

_{1}and

*Rt*

_{2}get the maximum values (i.e., 1) simultaneously, the array has the best performance of longitudinal-mode discrimination. To make

*Rt*maximum, following conditions should be satisfied, i.e.,

_{m}*A*

_{m}^{(−)}(

*L*) are induced by interference of two fields with the same intensity. Therefore,

*A*

_{m}^{(−)}(

*L*) (or

*f*) are most sensitive to the laser frequency (or longitudinal modes) at this time. Then, the array performs the longitudinal-mode discrimination best.

_{m}*ε*=

*ε*

_{1}=

*ε*

_{2},

*r*=

*r*

_{1}=

*r*

_{2}, and

*A*

_{1}

^{(+)}(

*L*) =

*A*

_{2}

^{(+)}(

*L*), we can get the relationship between

*ε*and

*r*from Eqs. (26). That is

*r*≠ 0,1, otherwise, the laser system is not an array of two mutually-injected lasers any more.

*L*̄

_{1}=2(

*L*+

*l*

_{1}

^{(c)}+

*l*

_{1}

^{(r)}+

*l*

_{1}'

^{(r)}) and

*L*̄

_{2}=2(

*L*+

*l*

_{2}

^{(c)}+

*l*

_{2}

^{(r)}+

*l*

_{2}'

^{(r)}). Here, ΔΦ

*is the phase difference per round-trip of the*

_{m}*m*th elementary laser. Of course, there are two independent Eqs. in Eqs. (28)–(30). Then, there should be some separated stationary modes satisfying these conditions within a given spectrum, as long as the bandwidth of the spectrum is broad enough. Furthermore, Eqs. (21) and (30) also indicate that increasing optical path differences (OPDs) of (

*l*−

_{cc}*l*

_{1}

^{(r)}) and (

*L*̄

_{2}−

*L*̄

_{1}) , will reduce the frequnecy spacing between adjacent stationary modes (corresponding to adjacent phase-locked states). Thus, there are two way to ensure the existence of phase-locking states in experiment, i.e., using broadband couplers and reflectors, and designing the compound cavity of this array reasonably to reducing the frequency spacing between adjacent phase-locked states.

## 4. Conclusion

## References and links

1. | T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. |

2. | Y. Li and D. Fan, “Beam combining of fiber laser,” Laser Optoelectron. |

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

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 |

5. | T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express |

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 |

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 |

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. |

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. |

10. | C. J. Corcoran and K. A. Pasch, “Modal analysis of a self-Fourier laser cavity,” J. Opt. A |

11. | C. J. Corcoran and F. Durville, “Experimental demonatration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. |

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 |

13. | T. Shirakawa, T. Saitou, Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express |

14. | H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, “200 W self-organized coherent fiber arrays,” in |

15. | H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. |

16. | S. Chen, Y. Li, K. Lu, and S. Zhou, “Efficient coherent combining of tunable erbium-doped fibre ring lasers,” J. Opt. A |

17. | M. Fridman, V. Eckhouse, N. Davidson, and A. Friesem, “Efficient coherent addition of fiber lasers in free space,” Opt. Lett. |

18. | B. Lei and Y. Feng, “Phase locking of an array of three fiber lasers by an all-fiber coupling loop,” Opt. Express |

19. | Z. Chen, J. Hou, P. Zhou, L. Liu, and Z. Jiang, “Mutual injection locking of two individual fiber lasers,” Acta Phys. Sin. |

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. |

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. |

22. | J. Cao, Q. Lu, J. Hou, and X. Xu, “Dynamical model for self-organized fiber laser arrays” Opt. Express |

23. | G. P. Agraval, |

**OCIS Codes**

(140.3290) Lasers and laser optics : Laser arrays

(140.3510) Lasers and laser optics : Lasers, fiber

(140.3520) Lasers and laser optics : Lasers, injection-locked

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 5, 2009

Revised Manuscript: April 9, 2009

Manuscript Accepted: April 13, 2009

Published: April 24, 2009

**Citation**

Jianqiu Cao, Qisheng Lu, Jing Hou, and Xiaojun Xu, "Self-organization of arrays of two mutually-injected fiber lasers: theoretical investigation," Opt. Express **17**, 7694-7701 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7694

Sort: Year | Journal | Reset

### References

- 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)
- 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. 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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). [CrossRef] [PubMed]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- G. P. Agraval, Applications of nonlinear fiber optics (Elsevier Science, USA, 2001).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.