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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 25379–25387
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Basic considerations on coherent combining of ultrashort laser pulses

Arno Klenke, Enrico Seise, Jens Limpert, and Andreas Tünnermann  »View Author Affiliations


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

The general setup for the coherent combination of N amplifiers is shown in Fig. 1
Fig. 1 Schematic setup of coherent addition of ultrashort laser pulses; Δφ: element for path length matching
. The ultrashort pulses from one mode-locked laser are split into N channels to start with mutually coherent pulses. These pulses are then amplified in their specific channel and finally recombined. An additional element has to be added in N-1 channels to match and stabilize the optical path lengths in the channels and ensure a constructive interference. Additionally, a stretcher and a compressor can be added to the system to create a chirped-pulse amplification (CPA) system. Assuming an identical behavior of all channels and an ideal combining element, the performance of the system could be in principle scaled by a factor of N compared to a single amplifier. However, when using ultrashort pulses, effects like dispersion and self-phase modulation (SPM), as well as optical path length differences (OPD) between the pulses, will affect the temporal and spectral phases of the combined pulses and, thus, lead to a non-perfect combination of the pulses. In order to characterize the performance of the combination process a figure of merit has been defined. In this paper, the dependency of this figure of merit on the effects mentioned above will be explored, under the assumption that the spectral intensity profiles of the pulses, as well as the spatial intensity and phase profiles of the beams are identical. An in-depth analysis of the detrimental effects caused by an imperfect overlap of the beams has already been published [10

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

The combining element plays an important role in the combining process, since it has to merge the beams coming from spatially separated amplification stages into one beam while preserving the beam quality and keeping losses as low as possible. Different elements have been successfully proposed such as partially reflective surfaces [11

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]

] or polarization dependent beam splitters [7

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]

]. Additionally, in the case of fiber lasers, fiber integrated couplers are another alternative. While all of these elements can in principle just combine two beams, by cascading them, the number of channels to be combined can be increased. Depending on the type of combining element, phase differences between the incoming pulses, e.g. OPDs, will have a different effect on the combined beam.

In the case of a partially reflective surface (Fig. 2a
Fig. 2 Combining process using (a) a partially reflective surface and (b) a polarization dependent cube.
), the power of the combined beam will depend on the constructive interference of the incident beams, while the rest of the power, i.e. that containing the non-interfering parts, will be emitted as a secondary beam in a different direction (see Fig. 2a). For every spectral component, the well-known interference formula can be used:
P(ω)=P0(ω)(1+cos(ΔΦ(ω)))
(1)
with P0(ω) 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 (Pcomb) and the secondary beam (Psecondary) can be measured and used in the following calculation for the visibility:

FOM=PcombPsecondaryPcomb+Psecondary
(2)

On the other hand, as shown in Fig. 2b, the combining process of two pulses can also be carried out using a polarization beam splitter. These elements are commercially available with excellent specifications, such as low losses and high damage thresholds. In this case, the two incident beams to be combined have to be one s- and the other p-polarized. This results in the combined beam being emitted exclusively from one port of the cube. Deviations from perfectly linearly polarized incident pulses and imperfections of the cube itself result in a power output at the so-called “dark port” and, therefore, in a reduction of the maximum achievable power of the combined beam. However, any spectral and/or temporal phase difference between the pulses will not be reflected in the power of the combined beam, but in its polarization state instead. Hence, a measurement of the polarization state is required to estimate the FOM of the combined beam.

Placing an analyzer at an angle of 45° behind the transmission port of the polarization beam splitter leads to an interference that can be expressed with Eq. (1). In this way the result is basically the same as for a partially reflective element. The definition of the figure of merit can also be used here by measuring the maximum and minimum average power obtained when rotating the analyzer. The figure of merit represents in this case the degree of linear polarization (DOLP) of the combined beam, and can be calculated as follows:

FOM=DOLP=PmaxPminPmax+Pmin
(3)

For two pulses with identical spectral intensities and assuming a perfect spatial overlap of the beams, the figure of merit will deliver a value between 0 and 1. Coherent interference results in strong modulations of the FOM for OPDs in the subwavelength region and there are local maxima with the periodicity equal to the wavelength. This resembles the behavior in the continuous wave regime. However, in the pulsed regime, the frequency dependence of the spectral phase differences has to be taken into account.

For a spectral phase difference 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 Pmax(ω) equals P(ω) and Pmin(ω) = 2P0(ω) - Pmax(ω). For every spectral component, the figure of merit can now be calculated by using Eq. (2). For 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 Pmax and Pmin and corresponds to the negative solution in the following calculation:

FOM(ω)=Pmax(ω)Pmin(ω)Pmax(ω)+Pmin(ω)=±2P0(ω)cos(ΔΦ(ω))2P0(ω)=±cos(ΔΦ(ω))
(4)

To calculate the FOM for the whole pulse, this result has to be weighted with the spectral intensity profile before being finally integrated over frequency:
FOM=Cs(ω)FOM(ω)dω=±Cs(ω)cos(ΔΦ(ω))dω
(5)
with the normalized spectral intensity s(ω) of the power P0(ω) and the normalization factor Cs(ω)dω=1. Please note that 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]

] in the case of polarization combining.

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

In this section the impact of OPDs between the pulses on the combining process is discussed.

An OPD between the two pulses is translated in a linear phase difference. The spectral phase shift of the OPD is thus given by:
ΔΦ(ω)=Δlc0ω
(6)
with the OPD Δl and the speed of light c0. As described in section 2, the interesting case is when Δl becomes a multiple of the wavelength λ0, 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 E1 and E2 and the common spectral intensity profile E0(ω):

E1(t)=12πE0(ω)eiΦchirp(ω)eiωtdωE(t)2=12πE0(ω)eiΦchirp(ω)eiΔΦdelay(ω)eiωtdωE(t)=12(E1(t)+E(t)2)
(7)

The pulses have the same chirp phase Φchirp(ω) and there is an additional spectral phase difference ΔΦdelay(ω) 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:

F~|E(t)|2dt=|E(ω)|2dω=12|E0(ω)|2|eiΦchirp(ω)+ei(Φchirp(ω)+ΔΦdelay(ω))|2dω=|E0(ω)|2(1+cos(ΔΦdelay(ω)))dω
(8)

The result shows that there is no dependency of the fluence of the combined pulse energy and therefore no dependency of the FOM on the chirp phase Φchirp(ω). For Gaussian pulses, the normalized spectral intensity is defined as:
s(ω)=e4ln2(ωω0ωFWHM)2
(9)
with the Full Width at Half Maximum (FWHM) bandwidth ωFWHM. By setting Δl=kλ0and using Eqs. (5), (6), (9), the FOM at these points can be analytically calculated:

FOM=Cs(ω)cos(ΔΦ(ω))dω=e(2πkωFWHMω0)216ln2
(10)

In Fig. 3
Fig. 3 Dependency of the local maxima of the FOM on a OPD for Gaussian pulses with a bandwidth up to 10nm at a typical center wavelength of 1030nm. The white line defines the boundary where the FOM falls below 95%.
, the result is depicted as a function of the OPD and of the spectral bandwidth. A center wavelength of 1030nm has been assumed to simulate a system working in the near infrared spectral region. The calculation has been done for bandwidths up to 10nm, which corresponds to pulses with transform limited pulse durations as short as ~150fs. A strong dependency of the acceptable delay on the signal bandwidth is immediately recognizable. For a system with a bandwidth of about 5nm, a OPD as large as 25 wavelengths (i.e. about 25µm in this case), is sufficient to keep the FOM above 90%. In comparison for shorter pulses (~10nm bandwidth) this value will drop to about 12µm. As can be seen from Eq. (10), the accuracy of the delay adjustment has to be increased by the same factor as the bandwidth to keep the FOM constant. It should be noted again that the OPD has to be stabilized to be as close as possible to one of the local maxima of the FOM. Only a variation of a fraction of wavelength is acceptable here.

4. Influences of SPM and dispersion on the combining process

ΔΦc(ω0)=Δlc0ω0=ΔBΔΦc(ω)=Δlc0ω=ΔBωω0=ΔB(1+ωω0ω0)ΔBfor|ωω0|<<ω0
(13)

While it is possible to calculate the FOM using a simulation with Eq. (2), it is also interesting to find analytical solutions. However, solving the integral in Eq. (5) proves to be difficult due to the cosine term. For small phase differences ΔΦ(ω) < π/4, a Taylor expansion of the cosine function up to the second order turns out to result in an error of less than 2%. By using Eq. (5),(9),(12),(13), the FOM can now be calculated:

FOM=Cs(ω)cos(ΔΦ(ω)+ΔΦc(ω))dω1+(1212123)ΔB23512ln(2)2ωFWHM4β22ΔL2+116ln(2)(1122)ωFWHM2β2ΔLΔB
(14)

So the quality of the combining process can be analytically calculated with just 4 parameters: ΔB, ΔL, the dispersion coefficient β2 and the spectral bandwidth ωFWHM. If we apply the condition 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<π4(1e1)11.2rad and ΔL<2πln2β2ωFWHM21.0m(for a bandwidth of 5nm)

The analytically calculated FOM depending on ΔB and ΔL is shown in Fig. 4
Fig. 4 Analytically calculated FOM for Gaussian pulses with a bandwidth of (a) 5nm, (b) 10nm, (c) 15nm propagating through fused silica, when a B-Integral or LDE difference is introduced.
. 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

To calculate the impact that fluctuations of the input power and amplification factor have on the FOM (only taking into account the resulting phase fluctuations because the influence of the power fluctuations itself can be neglected), one has to derive an expression for the corresponding change of the B-Integral. Assuming exponential amplification (unsaturated case), the formula for the B-Integral is [13

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

]:
B=CP0(g1)ln(g)1withC=8n2Lλ0MFD2frepτ
(15)
with the nonlinear coefficient n2, the propagation length L, the mode field diameter MFD, the repetition rate frep, the pulse duration τ, the input average power P0 and the amplification coefficient g. The change of the B-Integral depending on fluctuations of the input power and amplification coefficient (ΔP0, Δg) can then be approximated and a linear dependence on the absolute value of the B-Integral is found:

BP0ΔP0=BΔP0P0andBgΔgBΔggforg>>1
(16)

So while the FOM in Eq. (14) does not depend on the absolute value of B, for higher B-Integrals a relative fluctuation of the input power or the amplification coefficient will result in a larger change of the B-Integral and thus have a larger detrimental effect on the FOM.

This means that the sensitivity of a system will grow with higher B-Integrals, as shown in Fig. 5
Fig. 5 Dependency of the absolute change of the B-Integral and of the FOM on the value of the B-Integral, if a fluctuation of the input power and amplification coefficient of 5% is introduced.
. Hence, in a highly nonlinear regime, the stability of the amplifier input and output powers play a major role in determining the achievable FOM. In this case, additional stabilization of these factors might be required.

5. Figure of merit for more than two channels

So far, the presented formulas are limited to calculate the FOM for two channels. To calculate the FOM for a larger number of channels, the combined intensity has to be calculated first. This can be done in the same way for cascaded combining elements as if the combination process is realized in one step [10

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]

]. It is assumed that the combining fractions of the beam combining system are the same for every channel. Hence, the combined electric field for N channels with the phases Φn(ω) can be written as follows if equal intensities are assumed:

E(ω)=1Nn=1NE0(ω)eiΦn(ω)=E0(ω0)s(ω)Nn=1NeiΦn(ω)
(17)

The spectral intensity profile of the combined beam can now be calculated:
I(ω)~E*(ω)E(ω)=|E0(ω0)|2s(ω)N(N+i=1,ijNj=1Ncos(ΔΦij(ω)))
(18)
with the spectral phase differences between two channels ΔΦi,j(ω). Using Eq. (3), the FOM can be calculated for the system:
FOM=I(ω)(Ns(ω)I(ω))dωNs(ω)dω=CNs(ω)((2-N)+2Ni=1,ijNj=1Ncos(ΔΦij(ω)))dω=(2N1)+2N2i=1,ijNj=1NFOMij
(19)
with the FOM for two channels FOMij. In reality, the FOM between two random channels will result in approximately the same value FOM12, so the equation can be simplified further:

FOM(2N1)+2(11N)FOM12
(20)

6. Conclusion

In conclusion, we have investigated how different temporal and spectral effects impact a figure of merit introduced to characterize the coherent combination of two ultrashort pulses. These effects include SPM, dispersion and OPDs between the pulses. It has been shown that the detrimental effect of misalignments and fluctuations grow with increasing bandwidth of the pulses and with increasing B-Integral. However, even in those cases, an excellent FOM of > 90% should still be achievable with a carefully designed setup. This corresponds to a power loss of < 5% when the definition of the FOM is considered. In these cases, spatial effects, which were not considered in this paper, such as an imperfect overlap of the beams will have a larger impact on the combining process and might be the main factor in determining the viability of using the coherent combining approach. While the paper primarily deals with the combination of only two pulses, it is shown how the figure of merit can be calculated for a combining system with more than two channels.

Acknowledgements

This work has been partly supported by the German Federal Ministry of Education and Research (BMBF) and the European Research Council (ERC), SIRG 240460-PECS. A.K. acknowledges financial support by the Helmholtz-Institute Jena. E.S. acknowledges financial support by the Carl Zeiss Stiftung Germany.

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. 35(13), 2302–2304 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-13-2302. [CrossRef] [PubMed]

2.

P. Russbueldt, T. Mans, G. Rotarius, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “400W Yb:YAG Innoslab fs-Amplifier,” Opt. Express 17(15), 12230–12245 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12230. [CrossRef] [PubMed]

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. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94. [CrossRef] [PubMed]

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 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255. [CrossRef] [PubMed]

5.

R. Xiao, J. Hou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express 16(3), 2015–2022 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-2015. [CrossRef] [PubMed]

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 18(13), 13547–13553 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13547. [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]

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. 36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439. [CrossRef] [PubMed]

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. 35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052. [CrossRef] [PubMed]

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]

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]

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]

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 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.

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

  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.35(13), 2302–2304 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-13-2302 . [CrossRef] [PubMed]
  2. 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]
  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.35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 . [CrossRef] [PubMed]
  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. Express19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 . [CrossRef] [PubMed]
  5. 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]
  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. Express18(13), 13547–13553 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13547 . [CrossRef] [PubMed]
  7. 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]
  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.36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439 . [CrossRef] [PubMed]
  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.35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 . [CrossRef] [PubMed]
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