## Basic considerations on coherent combining of ultrashort laser pulses |

Optics Express, Vol. 19, Issue 25, pp. 25379-25387 (2011)

http://dx.doi.org/10.1364/OE.19.025379

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### Abstract

Coherent combining is a novel approach to scale the performance of laser amplifiers. The use of ultrashort pulses in a coherent combining setup results in new challenges compared to continuous wave operation or to pulses on the nanosecond timescale, because temporal and spectral effects such as self-phase modulation, dispersion and the optical path length difference between the pulses have to be considered. In this paper the impact of these effects on the combining process has been investigated and simple analytical equations for the evaluation of this impact have been obtained. These formulas provide design guidelines for laser systems using coherent combining. The results show that, in spite of the temporal and spectral effects mentioned above, for a carefully adjusted and stabilized system an excellent efficiency of the combining process can still be achieved.

© 2011 OSA

## 1. Introduction

10. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express **18**(24), 25403–25414 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403. [CrossRef] [PubMed]

## 2. General considerations on coherent beam combining of ultrashort pulses

11. L. Daniault, M. Hanna, L. Lombard, Y. Zaouter, E. Mottay, D. Goular, P. Bourdon, F. Druon, and P. Georges, “Coherent beam combining of two femtosecond fiber chirped-pulse amplifiers,” Opt. Lett. **36**(5), 621–623 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-621. [CrossRef] [PubMed]

7. E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express **18**(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827. [CrossRef] [PubMed]

*P*being the spectral power of the beams to be combined (they are assumed to have an identical beam profile) and ΔΦ(ω) being the spectral phase difference between them. In order to define a figure of merit (FOM) suitable for characterizing the combining process, the power in the combined beam (

_{0}(ω)*P*) and the secondary beam (

_{comb}*P*) can be measured and used in the following calculation for the visibility:

_{secondary}*ΔΦ(ω)*, the figure of merit can be analytically calculated at each frequency. Thus, as seen in Eq. (1), if

*cos(ΔΦ(ω))*≥ 0, then more power of this spectral component is in the combined beam than in the secondary beam. In this case, the maximum power of the combined beam

*P*equals

_{max}(ω)*P(ω)*and

*P*. For every spectral component, the figure of merit can now be calculated by using Eq. (2). For

_{min}(ω) = 2P_{0}(ω) - P_{max}(ω)*cos(ΔΦ(ω))*< 0, on the other hand, more power of this spectral component would instead be in the secondary beam. This results in a swap of the formula for

*P*and

_{max}*P*and corresponds to the negative solution in the following calculation:

_{min}*s(ω)*of the power

*P*and the normalization factor

_{0}(ω)*FOM(ω)*can be negative for some spectral components in this calculation. It is important to note that the sign in Eq. (4) has to be equal for all spectral components, and it should be chosen in such a way that the result of Eq. (4) is positive. Thus, if the function

*ΔΦ(ω)*can be calculated for a certain effect (e.g. SPM), it is now possible to estimate the FOM degradation that this effect causes. It is worth noting that the FOM is additionally a useful parameter to stabilize the system. In other words, the control loop can use the FOM as its feedback/error parameter and it can thus be locked to the best figure of merit. This can be done by using a phase modulation based locking system or with the Hänsch-Couillaud mechanism [12

12. T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. **35**(3), 441–444 (1980). [CrossRef]

## 3. Influence of optical path length differences on the combining process

*Δl*and the speed of light

*c*. As described in section 2, the interesting case is when

_{0}*Δl*becomes a multiple of the wavelength

*λ*, because in this case a local maximum for the combined power and, therefore, a local maximum for the FOM is reached. A decay of the FOM at these maxima is expected if the temporal overlap of the pulses is reduced. It should be noted that an OPD has the same impact for a chirped pulse as it would have for the corresponding transform limited pulse. This can be seen by interfering two pulses with the electric fields

_{0}*E*and

_{1}*E*and the common spectral intensity profile

_{2}*E*:

_{0}(ω)*Φ*and there is an additional spectral phase difference

_{chirp}(ω)*ΔΦ*for the OPD. The fluence of the combined pulse can now be calculated using Parseval’s theorem to investigate the impact of the chirp on the FOM:

_{delay}(ω)*Φ*For Gaussian pulses, the normalized spectral intensity is defined as:with the Full Width at Half Maximum (FWHM) bandwidth ω

_{chirp}(ω)._{FWHM}. By setting

## 4. Influences of SPM and dispersion on the combining process

_{0}[14

14. D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express **17**(7), 4997–5007 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4997. [CrossRef] [PubMed]

15. M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. **19**(24), 2149–2151 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-24-2149.

*L*and the second order dispersion coefficient

*β*. Hence, the spectral phase difference for the case of equal spectral intensities

_{2}*s(ω)*is:which only depends on the B-Integral difference

*ΔB*and LDE difference

*ΔL*of the two pulses. The parameter

*ΔL*is especially important for fiber lasers because of the large LDE typical of these devices. In reality, to achieve the best FOM, these phase differences will be partially compensated for by introducing an additional OPD between the pulses. This OPD is chosen to compensate the spectral phase difference for the center frequency:

*ΔB*,

*ΔL*, the dispersion coefficient

*β*and the spectral bandwidth

_{2}*ω*. If we apply the condition

_{FWHM}*ΔΦ(ω)*< π/4 for all the spectral components of

*s(ω)*with an intensity above 1/e, we can estimate the boundaries for the approximation:

*ΔB*and

*ΔL*is shown in Fig. 4 . Different bandwidths were chosen to show the dependence of the FOM on this parameter. The maximum deviation of the FOM between this analytically calculated solution and a simulation is 1%, which confirms the validity of the approximation taken to obtain Eq. (14). The graphs above show that for small deviations of the LDE, which should easily be realizable in a setup, a B-Integral difference

*ΔB*of 0.5 rad still results in an excellent value for the FOM of over 95%. This means that even at a high absolute B-Integral of ~10 rad, i.e. in a highly nonlinear regime, a good FOM can still be realistically reached. However, the dispersion term in the equation has a fourth order dependency on the bandwidth. Hence, with broad bandwidths the match of the LDE in the channels becomes critical.

### Influences of fluctuations of the input power and amplification coefficient

*n*, the propagation length

_{2}*L*, the mode field diameter

*MFD*, the repetition rate

*f*, the pulse duration

_{rep}*τ*, the input average power

*P*and the amplification coefficient

_{0}*g*. The change of the B-Integral depending on fluctuations of the input power and amplification coefficient (

*ΔP*,

_{0}*Δg*) can then be approximated and a linear dependence on the absolute value of the B-Integral is found:

## 5. Figure of merit for more than two channels

10. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express **18**(24), 25403–25414 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403. [CrossRef] [PubMed]

*Φ*

_{n}

*(ω)*can be written as follows if equal intensities are assumed:

*ΔΦ*

_{i,j}

*(ω)*. Using Eq. (3), the FOM can be calculated for the system:

*FOM*

_{ij}. In reality, the FOM between two random channels will result in approximately the same value

*FOM*

_{12}, so the equation can be simplified further:

## 6. Conclusion

## Acknowledgements

## References and links

1. | C. R. E. Baer, Ch. Kränkel, C. J. Saraceno, O. H. Heckl, M. Golling, R. Peters, K. Petermann, Th. Südmeyer, G. Huber, and U. Keller, “Femtosecond thin-disk laser with 141 W of average power,” Opt. Lett. |

2. | P. Russbueldt, T. Mans, G. Rotarius, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “400W Yb:YAG Innoslab fs-Amplifier,” Opt. Express |

3. | T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. |

4. | T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express |

5. | R. Xiao, J. Hou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express |

6. | R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express |

7. | E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express |

8. | E. Seise, A. Klenke, S. Breitkopf, M. Plötner, J. Limpert, and A. Tünnermann, “Coherently combined fiber laser system delivering 120 μJ femtosecond pulses,” Opt. Lett. |

9. | I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. |

10. | G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express |

11. | L. Daniault, M. Hanna, L. Lombard, Y. Zaouter, E. Mottay, D. Goular, P. Bourdon, F. Druon, and P. Georges, “Coherent beam combining of two femtosecond fiber chirped-pulse amplifiers,” Opt. Lett. |

12. | T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. |

13. | G. P. Agrawal, Nonlinear Fiber Optics 2nd Edition (Academic Press, 1995) |

14. | D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express |

15. | M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. |

**OCIS Codes**

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

(140.7090) Lasers and laser optics : Ultrafast lasers

(140.3298) Lasers and laser optics : Laser beam combining

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 18, 2011

Revised Manuscript: September 23, 2011

Manuscript Accepted: November 18, 2011

Published: November 28, 2011

**Citation**

Arno Klenke, Enrico Seise, Jens Limpert, and Andreas Tünnermann, "Basic considerations on coherent combining of ultrashort laser pulses," Opt. Express **19**, 25379-25387 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25379

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### References

- C. R. E. Baer, Ch. Kränkel, C. J. Saraceno, O. H. Heckl, M. Golling, R. Peters, K. Petermann, Th. Südmeyer, G. Huber, and U. Keller, “Femtosecond thin-disk laser with 141 W of average power,” Opt. Lett.35(13), 2302–2304 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-13-2302 . [CrossRef] [PubMed]
- P. Russbueldt, T. Mans, G. Rotarius, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “400W Yb:YAG Innoslab fs-Amplifier,” Opt. Express17(15), 12230–12245 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12230 . [CrossRef] [PubMed]
- T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett.35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 . [CrossRef] [PubMed]
- T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 . [CrossRef] [PubMed]
- R. Xiao, J. Hou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express16(3), 2015–2022 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-2015 . [CrossRef] [PubMed]
- R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express18(13), 13547–13553 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13547 . [CrossRef] [PubMed]
- E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express18(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827 . [CrossRef] [PubMed]
- E. Seise, A. Klenke, S. Breitkopf, M. Plötner, J. Limpert, and A. Tünnermann, “Coherently combined fiber laser system delivering 120 μJ femtosecond pulses,” Opt. Lett.36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439 . [CrossRef] [PubMed]
- I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett.35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 . [CrossRef] [PubMed]
- G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express18(24), 25403–25414 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403 . [CrossRef] [PubMed]
- L. Daniault, M. Hanna, L. Lombard, Y. Zaouter, E. Mottay, D. Goular, P. Bourdon, F. Druon, and P. Georges, “Coherent beam combining of two femtosecond fiber chirped-pulse amplifiers,” Opt. Lett.36(5), 621–623 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-621 . [CrossRef] [PubMed]
- T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun.35(3), 441–444 (1980). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics 2nd Edition (Academic Press, 1995)
- D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express17(7), 4997–5007 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4997 . [CrossRef] [PubMed]
- M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett.19(24), 2149–2151 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-24-2149 .

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