## Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays |

Optics Express, Vol. 18, Issue 25, pp. 25873-25886 (2010)

http://dx.doi.org/10.1364/OE.18.025873

Acrobat PDF (1539 KB)

### Abstract

We generalize the recently proposed model for coherent beam combining in passive fiber laser arrays [Opt. Express **17**, 19509 (2009)] to include the transient gain dynamics and the complication of counterpropagating waves, two important features characterizing actual experimental conditions. The extended model reveals that beam combining is not affected by the population relaxation process or the presence of backward propagating waves, which only serve to co-saturate the gain. The presence of nonresonant nonlinearity is found to reduce the coherent combining efficiency at high power levels. We show that the array lases at the frequencies with minimum overall losses when multiple loss mechanisms are present.

© 2010 OSA

## 1. Introduction

1. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. **30**(11), 1339–1341 (2005). [CrossRef] [PubMed]

5. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express **10**(21), 1167–1172 (2002). [PubMed]

6. J. Q. Cao, J. Hou, Q. S. Lu, and X. J. Xu, “Numerical research on self-organized coherent fiber laser arrays with circulating field theory,” J. Opt. Soc. Am. B **25**(7), 1187–1192 (2008). [CrossRef]

10. K. Wiesenfeld, S. Peles, and J. L. Rogers, “Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization,” IEEE J. Sel. Top. Quantum Electron. **15**(2), 312–319 (2009). [CrossRef]

11. T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Model for passive coherent beam combining in fiber laser arrays,” Opt. Express **17**(22), 19509–19518 (2009). [CrossRef] [PubMed]

## 2. Model

*z =*0, excite active ions and give rise to gain at longer wavelengths. The fiber Bragg gratings (FBG) provide ~100% feedback at the right ends of the fibers, while differential power reflectivities, R

_{1}and R

_{2}in Fig. 1, are applied to the output ports of the 50:50 coupler at the left-hand sides. Assuming single polarization, the coherent waves propagating in + z and −z directions in each fiber laser are governed by the nonlinear Schrödinger equation together with the rate equation [12, 13]:

*j*= 1) and second (

*j*= 2) fiber respectively. Various effects including linear gain

*g*, fiber losses

_{j}*α*, the inverse of the group velocity

*β*, the frequency-dependent losses

_{1}*b*, the group velocity dispersion

*β*, and the nonresonant Kerr nonlinearity

_{2}*γ*are all incorporated in Eqs. (1) and (2). In Eq. (3) for gain dynamics,

*g*specifies the unsaturated gain while the second and third terms respectively describe the process of excited population relaxation with upper-state lifetime

_{0j}*τ*and laser gain saturation at high intensity fields. We normalize the electric field amplitudes so

15. V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express **13**(18), 6791–6797 (2005). [CrossRef] [PubMed]

*t*within the computation window. We handle this complication by using an iterative SSFM [16

16. D. Hollenbeck and C. D. Cantrell, “Parallelizable, bidirectional method for simulating optical-signal propagation,” J. Lightwave Technol. **27**(12), 2140–2149 (2009). [CrossRef]

*g*abruptly as assumed in many cases [15

_{j}15. V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express **13**(18), 6791–6797 (2005). [CrossRef] [PubMed]

17. J. Kanka, “Numerical simulation of subpicosecond soliton formation in a nonlinear coupler laser,” Opt. Lett. **19**(22), 1873–1875 (1994). [CrossRef] [PubMed]

18. E. Martí-Panameño, L. C. Gomez-Pavon, A. Luis-Ramos, M. M. Méndez-Otero, and M. D. I. Castillo, “Self-mode-locking action in a dual-core ring fiber laser,” Opt. Commun. **194**, 409–414 (2001). [CrossRef]

9. J. L. Rogers, S. Peles, and K. Wiesenfeld, “Model for high-gain fiber laser arrays,” IEEE J. Quantum Electron. **41**(6), 767–773 (2005). [CrossRef]

*Δt*to be four times the roundtrip duration in the simulation since numerical accuracy is ensured when

*Δt*(~1 μs) is much smaller than the population relaxation time constant

*τ*(10 ms). To validate our derivation of the dynamical model, an example is given for a two-channel fiber laser array of fiber lengths 24.3 and 24.0 m. Here by setting

*γ =*0 W

^{−1}m

^{−1}the array is assumed to be linear so the nonlinear phases will not modify the distinct modes of the laser cavity [11

11. T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Model for passive coherent beam combining in fiber laser arrays,” Opt. Express **17**(22), 19509–19518 (2009). [CrossRef] [PubMed]

*rtstps*= 70.) Other parameter values used are

## 3. Simulation results

*L*= 24.3 m) at steady state. It is clear that the backward signal dominates in this efficient backward pumping configuration [19

_{1}19. B. N. Upadhyaya, U. Chakravarty, A. Kuruvilla, A. K. Nath, M. R. Shenoy, and K. Thyagarajan, “Effect of steady-state conditions on self-pulsing characteristics of Yb-doped cw fiber lasers,” Opt. Commun. **281**(1), 146–153 (2008). [CrossRef]

^{−1}and these equal the roundtrip losses of

*R*= 0.04) port, while a negligible amount leaks through the lower, angle-cleaved (

_{1}*R*= 0) one. Taking

_{2}*N*- channel array as

20. D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, ““Coherent combining of two Nd: YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B-Lasers. **75**, 503–507 (2002). [CrossRef]

11. T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Model for passive coherent beam combining in fiber laser arrays,” Opt. Express **17**(22), 19509–19518 (2009). [CrossRef] [PubMed]

*z*= 0 before the 50:50 coupler, where

## 4. Nonlinearity

*n*[21–23

_{2}23. B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. **34**(7), 863–865 (2009). [CrossRef] [PubMed]

23. B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. **34**(7), 863–865 (2009). [CrossRef] [PubMed]

*n*has no apparent effect on the combining efficiency in a two-channel fiber laser array [11

_{2}**17**(22), 19509–19518 (2009). [CrossRef] [PubMed]

*γ*by more than two orders of magnitude and thereby force the effects of nonlinearity to be manifested at much lower power levels. Assuming

*γ*to be 0.9 W

^{−1}m

^{−1}, the previous simulation is repeated without changing other parameters. The two array outputs are plotted in Fig. 5(a) and 5(b) respectively for both temporal (left) and spectral (right) domains. Compared to Fig. 3, several apparent differences can be observed. Firstly, the FWHMs of each spectral packet broaden considerably. Second, in contrast to the centered power spectrum of the linear arrays, the nonlinearity causes the frequency components to spread and so the outermost packets are most intense. The resultant spectrum is not constrained to the parabolic loss profile and looks similar to that of pulse propagation in the presence of self-phase modulation. Third, the combining efficiency reduces to 85.5% with a significant amount of power leaking from the lossy port. The final comparison is made to the phase spectrum of the circled spectral packet in Fig. 5(a) and its linear counterpart of Fig. 3(a). The relative phase difference plots show that the range of

*P*. On the other hand, when

_{1}*P*decreases and more power emerges out of the lower, angle-cleaved port as is evident in Fig. 5. The drop in combining efficiency is thus seen to be a result of the increasing bandwidth of the power spectrum, in particular, the broadening of each spectral packet under modulation.

_{1}## 5. Array lasing frequencies - the minimum loss

**17**(22), 19509–19518 (2009). [CrossRef] [PubMed]

*b*may give rise to reduced combining efficiency as well as shifted lasing frequencies in a two-channel fiber laser array. It is essential to understand how the resonant frequencies are determined and why the frequency shift occurs in the presence of additional loss sources. A reasonable expectation is that the array chooses to lase at the frequency that experiences the least overall losses. In order to verify this point, we present a simple loss analysis based on the unidirectional two-channel fiber laser array. The ring cavity configuration is adopted here as seen in Fig. 7 since it is simpler and we have shown earlier that the coherent beam combining is not affected by the backward propagating waves of standing-wave cavities.

*P*at the reference plane just before the coupler, so we can writewhere

*R*is the power reflectivity,

*g*is the saturated gain,

*α*is the linear loss and

*b*is the loss dispersion coefficient. We take the logarithm of Eq. (10) and the expression for the loss is given as the right hand side of Eq. (11).The frequency dependent loss profile can thus be readily plotted by plugging in

*b*= 0 ps

^{2}m

^{−1}and (b)

*b*= 0.13 ps

^{2}m

^{−1}respectively. (The very small length difference is chosen to ensure the spike separation and the frequency shifts are large enough for clear visualization.) It is clear the combining efficiency drops considerably and the lasing frequency shifts from 126.5 GHz to 45.79 GHz in the presence of nonzero loss dispersion. Utilizing Eq. (11), we plot the frequency-dependent loss on a logarithmic scale with blue solid lines and further overlap them with the output lasing frequencies (red solid lines) in Fig. 9 for better visualization. The loss curves exhibit minimum values near −300 GHz and 100 GHz in Fig. 9(a) and around 50 GHz in Fig. 9(b). In both cases, the good agreement between array resonant modes and the location of the minimum losses validates the hypothesis that the coupled array finds the mode with minimum overall losses.

## 6. Conclusion

*n*induces spectral broadening and reduces the combining efficiency at high power levels. We explore the working principle of the array and demonstrate that it is based on the selection of composite cavity modes with the minimum overall losses.

_{2}## Appendix

## Array mode spacing – the greatest common divisor

*N*coupled lasers the separation between the maxima of the modulation envelope can be found by examining the condition for constructive interference at all the 50:50 couplers. Since lasing of the composite cavity should occur near these maxima (corresponding to frequencies of minimum loss) this analysis helps to make sense of the complicated spectra observed in multi-element arrays.

_{1}and the upper output ports of M

_{2}and M

_{3}respectively when

*n*is refractive index of the fiber and

_{1}_{3}and it depends only on the fiber lengths

*L*and

_{2}*L*. This can be understood by calculating the phase of the output fields from the coupler M

_{3}_{1}by

_{1}such that

_{2}with the result of

_{3}are merely characterized by fiber lengths

*L*and

_{2}*L*in Eq. (A2).

_{3}*L*to

_{1}*L*, the exact solution

_{4}*f*generally does not exist for all three equations in Eq. (A1) even with the degrees of freedom provided by

*k*= 1⋯3, indicates the deviations of

**17**(22), 19509–19518 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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

2. | W. Z. Chang, T. W. Wu, H. G. Winful, and A. Galvanauskas, “Array size scalability of passively coherently phased fiber laser arrays,” Opt. Express |

3. | 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 |

4. | A. Shirakawa, K. Matsuo, and K. Ueda, “Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control,” in Conference on Fiber Lasers II, L. N. Durvasula, A. J. W. Brown, and J. Nilsson, eds. (San Jose, CA, 2005), pp. 165–174. |

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

6. | J. Q. Cao, J. Hou, Q. S. Lu, and X. J. Xu, “Numerical research on self-organized coherent fiber laser arrays with circulating field theory,” J. Opt. Soc. Am. B |

7. | D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. |

8. | W. Ray, J. L. Rogers, and K. Wiesenfeld, “Coherence between two coupled lasers from a dynamics perspective,” Opt. Express |

9. | J. L. Rogers, S. Peles, and K. Wiesenfeld, “Model for high-gain fiber laser arrays,” IEEE J. Quantum Electron. |

10. | K. Wiesenfeld, S. Peles, and J. L. Rogers, “Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization,” IEEE J. Sel. Top. Quantum Electron. |

11. | T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Model for passive coherent beam combining in fiber laser arrays,” Opt. Express |

12. | G. P. Agrawal, |

13. | A. E. Siegman, |

14. | A. Yariv, |

15. | V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express |

16. | D. Hollenbeck and C. D. Cantrell, “Parallelizable, bidirectional method for simulating optical-signal propagation,” J. Lightwave Technol. |

17. | J. Kanka, “Numerical simulation of subpicosecond soliton formation in a nonlinear coupler laser,” Opt. Lett. |

18. | E. Martí-Panameño, L. C. Gomez-Pavon, A. Luis-Ramos, M. M. Méndez-Otero, and M. D. I. Castillo, “Self-mode-locking action in a dual-core ring fiber laser,” Opt. Commun. |

19. | B. N. Upadhyaya, U. Chakravarty, A. Kuruvilla, A. K. Nath, M. R. Shenoy, and K. Thyagarajan, “Effect of steady-state conditions on self-pulsing characteristics of Yb-doped cw fiber lasers,” Opt. Commun. |

20. | D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, ““Coherent combining of two Nd: YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B-Lasers. |

21. | E. J. Bochove, “Effect of nonlinear phase on the passive phase locking of an array of fiber lasers of random lengths,” in |

22. | C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. |

23. | B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. |

24. | H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, “200 W self-organized coherent fiber arrays,” in 2005 Conference on Lasers & Electro-Optics (2005), pp. 532–534. |

25. | A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished at http://www.stanford.edu/~siegman/Coupled%20Fiber%20Lasers/coupled_fiber_modes.pdf (2004). |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(140.3290) Lasers and laser optics : Laser arrays

(140.3510) Lasers and laser optics : Lasers, fiber

(140.3298) Lasers and laser optics : Laser beam combining

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 29, 2010

Revised Manuscript: November 8, 2010

Manuscript Accepted: November 11, 2010

Published: November 25, 2010

**Citation**

Tsai-wei Wu, Wei-zung Chang, Almantas Galvanauskas, and Herbert G. Winful, "Dynamical, bidirectional model for coherent beam combining in passive fiber laser arrays," Opt. Express **18**, 25873-25886 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25873

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### References

- H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339–1341 (2005). [CrossRef] [PubMed]
- W. Z. Chang, T. W. Wu, H. G. Winful, and A. Galvanauskas, “Array size scalability of passively coherently phased fiber laser arrays,” Opt. Express 18(9), 9634–9642 (2010). [CrossRef] [PubMed]
- 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(2), 87–97 (2003). [CrossRef] [PubMed]
- A. Shirakawa, K. Matsuo, and K. Ueda, “Fiber laser coherent array for power scaling, bandwidth narrowing, and coherent beam direction control,” in Conference on Fiber Lasers II, L. N. Durvasula, A. J. W. Brown, and J. Nilsson, eds. (San Jose, CA, 2005), pp. 165–174.
- A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [PubMed]
- J. Q. Cao, J. Hou, Q. S. Lu, and X. J. Xu, “Numerical research on self-organized coherent fiber laser arrays with circulating field theory,” J. Opt. Soc. Am. B 25(7), 1187–1192 (2008). [CrossRef]
- D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. 12(6), 445–447 (2005). [CrossRef]
- W. Ray, J. L. Rogers, and K. Wiesenfeld, “Coherence between two coupled lasers from a dynamics perspective,” Opt. Express 17(11), 9357–9368 (2009). [CrossRef] [PubMed]
- J. L. Rogers, S. Peles, and K. Wiesenfeld, “Model for high-gain fiber laser arrays,” IEEE J. Quantum Electron. 41(6), 767–773 (2005). [CrossRef]
- K. Wiesenfeld, S. Peles, and J. L. Rogers, “Effect of Gain-Dependent Phase Shift on Fiber Laser Synchronization,” IEEE J. Sel. Top. Quantum Electron. 15(2), 312–319 (2009). [CrossRef]
- T. W. Wu, W. Z. Chang, A. Galvanauskas, and H. G. Winful, “Model for passive coherent beam combining in fiber laser arrays,” Opt. Express 17(22), 19509–19518 (2009). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear fiber optics, third edition (Academic Press, 2001), pp. 263–264.
- A. E. Siegman, Lasers (University Science Books (January 1986)), pp. 485–487.
- A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th Edition (Oxford University Press, USA, 2007), pp. 563–565.
- V. Roy, M. Piché, F. Babin, and G. W. Schinn, “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser,” Opt. Express 13(18), 6791–6797 (2005). [CrossRef] [PubMed]
- D. Hollenbeck and C. D. Cantrell, “Parallelizable, bidirectional method for simulating optical-signal propagation,” J. Lightwave Technol. 27(12), 2140–2149 (2009). [CrossRef]
- J. Kanka, “Numerical simulation of subpicosecond soliton formation in a nonlinear coupler laser,” Opt. Lett. 19(22), 1873–1875 (1994). [CrossRef] [PubMed]
- E. Martı́-Panameño, L. C. Gomez-Pavon, A. Luis-Ramos, M. M. Méndez-Otero, and M. D. I. Castillo, “Self-mode-locking action in a dual-core ring fiber laser,” Opt. Commun. 194, 409–414 (2001). [CrossRef]
- B. N. Upadhyaya, U. Chakravarty, A. Kuruvilla, A. K. Nath, M. R. Shenoy, and K. Thyagarajan, “Effect of steady-state conditions on self-pulsing characteristics of Yb-doped cw fiber lasers,” Opt. Commun. 281(1), 146–153 (2008). [CrossRef]
- D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, ““Coherent combining of two Nd: YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B-Lasers. 75, 503–507 (2002). [CrossRef]
- E. J. Bochove, “Effect of nonlinear phase on the passive phase locking of an array of fiber lasers of random lengths,” in Integrated Photonics and Nanophotonics Research and Applications (IPNRA) (Honolulu, Hawaii, 2009).
- C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007). [CrossRef]
- B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009). [CrossRef] [PubMed]
- H. Bruesselbach, M. Minden, J. L. Rogers, D. C. Jones, and M. S. Mangir, “200 W self-organized coherent fiber arrays,” in 2005 Conference on Lasers & Electro-Optics (2005), pp. 532–534.
- A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished at http://www.stanford.edu/~siegman/Coupled%20Fiber%20Lasers/coupled_fiber_modes.pdf (2004).

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