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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 12197–12203
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Amplification of ultra-short optical pulses in a two-pump fiber optical parametric chirped pulse amplifier

Arnaud Mussot, Alexandre Kudlinski, Patrick Beaure d’Augères, and Emmanuel Hugonnot  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 12197-12203 (2013)
http://dx.doi.org/10.1364/OE.21.012197


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Abstract

We demonstrate with realistic numerical simulations that fiber optical parametric chirped pulse amplification is able to amplify ultra-short optical pulses. Such amplifiers driven by two-pump waves can amplify pulse bandwidth twice as large as the one of a single pump configuration. We show that pulses as short as 50 fs can be directly amplified. In addition, we take benefit from the saturation regime to achieve spectral broadening which makes possible to reduce pulse duration down to 15 fs.

© 2013 OSA

1. Introduction

In this paper, we numerically demonstrate that the whole gain band can be exploited in a two-pump FOPCPA configuration by launching a chirped signal in the middle of the pumps. Moreover, we show that spectral broadening originating from the saturation regime makes it possible to compress amplified pulses down to 15 fs in a realistic all-fiber device.

2. Model

Our numerical simulations are performed by integrating the extended nonlinear Schrödinger equation [13

13. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

]:
E(z,τ)z=iβ222E(z,τ)τ2+β363E(z,τ)τ3+iβ4244E(z,τ)τ4+iγ|E|2E(z,τ)+iγhR(t)|E(z,ττ')|2dτ'E(z,τ)
(1)
with β2,3,4 the second, third and fourth order dispersion terms, γ the nonlinear coefficient and hR(t) the Raman response deduced from experimental measurements. It has been numerically integrated by using an adaptative split step method [14

14. O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,” J. Lightwave Technol. 21(1), 61–68 (2003). [CrossRef]

], with a precision of 10−8, 1021 points over 60 THz which leads to a temporal resolution of 4.15 fs. One photon per mode has been added on the input signal in order to account for quantum fluctuations. We can note that this is the same equation used in Ref [9

9. D. Bigourd, L. Lago, A. Kudlinski, E. Hugonnot, and A. Mussot, “Dynamics of fiber optical parametric chirped pulse amplifiers,” J. Opt. Soc. Am. B 28(11), 2848–2854 (2011). [CrossRef]

], where the good agreement with experiments confirms that it accurately models these kinds of setups. The signal is a Gaussian pulse centered at λC = 1065 nm with a duration of 50 fs at full width at half maximum (FWHM) and an energy of 1 pJ. Such ultra-short pulses are now available with ytterbium fiber oscillator technology [11

11. T. Kurita, H. Yoshida, T. Kawashima, and N. Miyanaga, “Generation of sub-7-cycle optical pulses from a mode-locked ytterbium-doped single-mode fiber oscillator pumped by polarization-combined 915 nm laser diodes,” Opt. Lett. 37(19), 3972–3974 (2012). [CrossRef] [PubMed]

]. It is then stretched to a duration of 4.5 ns at −20 dB (1.74 ns at - 3 dB) with a purely quadratic stretcher (for instance, a very striking option for a compact stretcher is to use long chirped fiber Bragg grating [15

15. P. C. Chou, H. A. Haus, and J. F. Brennan III, “Reconfigurable time-domain spectral shaping of an optical pulse stretched by a fiber Bragg grating,” Opt. Lett. 25(8), 524–526 (2000). [CrossRef] [PubMed]

]). It is coupled into an optical fiber serving as the parametric gain medium together with two pumps. These pumps are square pulses of 6 ns duration at −20 dB with a peak power of 200 W and are frequency shifted of +/−11 THz from the central frequency, i.e. at 1025 and 1108 nm respectively. Note that these wavelengths are located within the ytterbium gain band and can then easily be generated by using standard ytterbium-doped fiber amplifiers [16

16. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]

]. The input overall spectrum is represented in red line in Fig. 1(a)
Fig. 1 (a) Input spectrum (red line) and small signal gain curve (dotted black line) calculated from a numerical integration of the nonlinear Schrödinger equation (Eq. (1)). (b) Input temporal characteristics normalized to unity, stretched signal in red and pump in blue.
.

The input temporal profiles of the pump and stretched signal are displayed in blue and red lines respectively in Fig. 1(b). The fiber is an air/silica microstructured optical fiber with the following realistic parameters: β2C) = −1.05 × 10−29 s2/m, β3C) = 0.6 × 10−40 s3/m, β4C) = −1 × 10−55 s4/m, γ = 9 /W/km, fiber length L = 2 m, linear attenuation α = 13.5 dB/km and the zero dispersion wavelength is 1064.9 nm. This fiber is similar to the one used in the experimental work presented in Ref [9

9. D. Bigourd, L. Lago, A. Kudlinski, E. Hugonnot, and A. Mussot, “Dynamics of fiber optical parametric chirped pulse amplifiers,” J. Opt. Soc. Am. B 28(11), 2848–2854 (2011). [CrossRef]

]. The small signal gain curve calculated by numerically integrating the nonlinear Schrodinger equation [1

1. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, 1st ed. (Cambridge University Press, 2007).

] is represented in black dots in Fig. 1(a). It shows that more than 80 nm bandwidth (central region) can be achieved with a 40 dB maximum amplification gain.

3. Results

4.1 Utilization of the whole bandwidth of the amplifier

At the output of the amplifier, the signal was extracted by means of a spectral filter to remove the pumps (whose characteristics are similar to the ones used for the spectrogram) and then, in a second step it was perfectly recompressed. This includes a perfect compensation of the whole set-up dispersion up to fourth order. This is not unrealistic since, experimentally, besides compressor (using either diffraction gratings or Bragg gratings), one has to properly manage dispersion via a careful design of a dispersion compensating fiber [18

18. L. Grüner-Nielsen, D. Jakobsen, K. G. Jespersen, and B. Pálsdóttir, “A stretcher fiber for use in fs chirped pulse Yb amplifiers,” Opt. Express 18(4), 3768–3773 (2010). [CrossRef] [PubMed]

] and/or via active spectral phase shaping with either a spatial light modulator [19

19. J. Prawiharjo, N. K. Daga, R. Geng, J. H. Price, D. C. Hanna, D. J. Richardson, and D. P. Shepherd, “High fidelity femtosecond pulses from an ultrafast fiber laser system via adaptive amplitude and phase pre-shaping,” Opt. Express 16(19), 15074–15089 (2008). [CrossRef] [PubMed]

] or an acousto-optic programmable filter [20

20. D. N. Papadopoulos, I. Martial, M. Hanna, F. Druon, and P. Georges, “Active spectral phase control by use of an acousto-optic programmable filter in high-repetition-rate sub-80 fs nonlinear fiber amplifiers,” Opt. Lett. 33(13), 1431–1433 (2008). [CrossRef] [PubMed]

] or an electro-optic phase modulator [21

21. J. van Howe, G. Zhu, and C. Xu, “Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems,” Opt. Lett. 31(11), 1756–1758 (2006). [CrossRef] [PubMed]

] to achieve such perfect compensation.

The spectrogram illustrating this recompression process is represented in Fig. 3(a)
Fig. 3 (a) Spectrogram of the recompressed signal after parametric amplification. (b) Temporal traces of the recompressed signal at L = 2 m (black dash-dot line) and at L = 4 m (blue solid line)) compared to the initial signal before stretching (red dashed line), normalized to unity. Insets: temporal profile of the recompressed signal in log scale (left inset) and over a larger temporal span (right inset). Data correspond to L = 2 m.
. The “X” trace evolves into a strait vertical line with a slanting component. As only half of the spectro-temporal points are concerned by the phase law of the compressor, the spectrogram trace can be divided into two parts. The vertical trace corresponds to spectro-temporal components of the signal for which the chirp has been quasi-perfectly cancelled by the linear recompression process. Indeed, we do not obtain a perfect recompression due to gain narrowing process because the bandwidth of the amplifier, defined in the small signal regime, is shorter than the FWHM of the signal. This is clearly illustrated in Fig. 1(a) where input signal spectrum and gain curve are superimposed. The oblique part of the spectrogram corresponds to the spectro-temporal components of the idler wave exhibiting a chirp which is now twice larger than the one of the input signal. Indeed, we add a chirp which is of the same sign than the one of the input trace. Note that it is quite possible to alternatively compress either the spectro-temporal components of the signal or of the idler by modifying the sign of the dispersion of the compressor. A close-up of the recompressed signal is represented in Fig. 3(b) in black dash-dot line and for comparison the input signal has been superimposed in red dashed lines. The duration of the amplified signal after recompression is 90 fs which is reasonably close to the input duration of 50 fs. Another important issue concerns the contrast of the amplified signal which is of primary importance for laser matter experiments [22

22. J. W. Dawson, M. J. Messerly, H. H. Phan, J. K. Crane, R. J. Beach, C. W. Siders, and C. Barty, “High-Energy, Short-Pulse Fiber Injection Lasers at Lawrence Livermore National Laboratory,” IEEE J. Sel. Top. Quantum Electron. 15(1), 207–219 (2009). [CrossRef]

]. In fact, it is one of the main drawbacks of ytterbium fiber amplification where good temporal quality is obtained only at the expense of temporal cleaning systems with high losses, instability, increased complexity and loss of compactness [23

23. Y. Zaouter, L. P. Ramirez, D. N. Papadopoulos, C. Hönninger, M. Hanna, F. Druon, E. Mottay, and P. Georges, “Temporal cleaning of a high-energy fiber-based ultrafast laser using cross-polarized wave generation,” Opt. Lett. 36(10), 1830–1832 (2011). [CrossRef] [PubMed]

]. Insets in Fig. 3(b) represent the output signal in logarithmic scales over large temporal windows of 2 ps and 100 ps respectively. We can see that a quite good temporal contrast of 26 dB can be achieved in the ps range and more than 42 dB in the hundred ps range. The slight degradation of the quality of amplification is due to all spectro-temporal components located in the slanting branch of the “X” trace after recompression which are located at the feet of the pulse, without significant gain narrowing effect.

4.2 Saturation to reach the gain broadening effect

On the same figure, the FWHM duration of the recompressed pulse is superimposed in blue circles. In the linear regime, due to the gain narrowing process, the signal duration increases to reach a maximum value of 90 fs at L = 2m (see the signal shape in Fig. 3(b) in black dashed-dotted line). Figures 4(b) and 4(c) clearly illustrate that the spectrum of the signal in between the pumps at 1.2 m and 2 m (blue curves) is thinner than the input signal spectrum which is superimposed in red lines for the sake of clarity. When the amplifier starts to saturate, this leads to a lowering of the gain per unit length, i.e. the gain in energy is no longer exponential and in addition, the saturation leads to a distortion of the spectral gain curve. This can be seen in Fig. 4(d) where the power of spectral components located in the wings of the signal grows more rapidly that the central part of the signal. This process leads to a gain broadening mechanism and to the shortening of the signal duration. As a consequence, the signal FWHM evolves from 90 fs at L = 2 m to 52 fs at L = 4 m. This value is very close to the initial signal duration of 50 fs, showing that these kinds of setups are capable of amplifying such ultra-short pulses.

4.3 Impact of the filter bandwidth

As it can be seen, the pulse is reshaped and by increasing the filter bandwidth, pulses as short as 15 fs are generated. It means that most of the spectral components of the output spectrum of such saturated two pump amplifier follow an appropriate phase law that allows to achieve pulse shortening after the recompression stage. The limit of this shortening effect is fixed by the spectral acceptance of all the optical elements (amplifier, stretcher…) and by the relative level of the signal wings compared to the noise floor. Then, in order to get a complete insight of the impact of the filter bandwidth, we focused our investigations on the contrast of these signals. To this aim, Fig. 5(b) represents temporal signals with a scale in dBm. It shows that the use of filters broader than the pump spacing leads to a contrast degradation of a few dB for the shorter pulse (42 dB to 37 dB). However, this value remains still by far better than ytterbium-doped fiber amplifiers.

4. Conclusion

We numerically demonstrate with realistic numerical simulations that ultra-short optical pulses can be amplified with FOPCPA systems. We show that the available bandwidth of FOPCPAs can be twice as large as in standard setups by using a two-pump configuration and a short chirped signal launched exactly in between them. The amplifier operates in the phase insensitive regime since each generated idlers are temporally separated from signals. This is due to a combination of the frequency chirp of the signal and to the phase conjugation properties of idlers waves in parametric amplifiers. In addition, the gain narrowing process is counterbalanced by the amplifier saturation leading to spectral broadening. As a result, ultra-short pulses can be amplified in such devices. As an example, we show that a launched signal of only 50 fs can be amplified by more than 57 dB with a pretty good signal contrast of 42 dB in the hundreds of ps range. Moreover, at the expense of slight contrast degradation due to the residual pump, we show that we can take advantage of the spectral broadening associated to the use of spectral filters broader than the pump frequency shift to shorten input pulses. In this example, by launching 50 fs pulses we obtain pulses as short as 15 fs. Such a result paves the way toward all-fiber generation and amplification of few cycle optical pulse [24

24. A. Mussot, A. Kudlinski, and E. Hugonnot, “Procédé and dispositif d’amplification paramétrique optique d’impulsions à dérive en fréquence, utilisant deux signaux de pompe and permettant l’élargissement de la bande spectrale de gain,” U.S. Patent FR 11 61642 (dec 14, 2011).

]. This numerical study highlights the benefit of a two-pump configuration for the amplification of ultra-short pulses compared to ytterbium fiber systems where gain narrowing strongly increases the signal duration.

Acknowledgments

This work was partly supported by the Agence Nationale de la Recherche through the ANR FOPAFE project, by the French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and Fonds Européen de Développement Régional through the “Contrat de Projets Etat Région (CPER) 2007-2013” and the “Campus Intelligence Ambiante (CIA)”, the equipex FLUX and the labex CEMPI.

References and links

1.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, 1st ed. (Cambridge University Press, 2007).

2.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002). [CrossRef]

3.

S. Radic, “Parametric Signal Processing,” IEEE J. Sel. Top. Quantum Electron. 18(2), 670–680 (2012). [CrossRef]

4.

A. Vedadi, M. Jamshidifar, and M. E. Marhic, “Continuous-wave one-pump fiber optical parametric amplifier with 230 nm gain bandwidth,” paper 1.1.4 in 35th European Conference on Optical Communication (ECOC), p. 1–2 (2009).

5.

R. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11(3), 100–103 (1975). [CrossRef]

6.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8(3), 538–547 (2002). [CrossRef]

7.

C. Caucheteur, D. Bigourd, E. Hugonnot, P. Szriftgiser, A. Kudlinski, M. Gonzalez-Herraez, and A. Mussot, “Experimental demonstration of optical parametric chirped pulse amplification in optical fiber,” Opt. Lett. 35(11), 1786–1788 (2010). [CrossRef] [PubMed]

8.

D. Bigourd, L. Lago, A. Mussot, A. Kudlinski, J.-F. Gleyze, and E. Hugonnot, “High-gain fiber, optical-parametric, chirped-pulse amplification of femtosecond pulses at 1 μm,” Opt. Lett. 35(20), 3480–3482 (2010). [CrossRef] [PubMed]

9.

D. Bigourd, L. Lago, A. Kudlinski, E. Hugonnot, and A. Mussot, “Dynamics of fiber optical parametric chirped pulse amplifiers,” J. Opt. Soc. Am. B 28(11), 2848–2854 (2011). [CrossRef]

10.

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). [CrossRef] [PubMed]

11.

T. Kurita, H. Yoshida, T. Kawashima, and N. Miyanaga, “Generation of sub-7-cycle optical pulses from a mode-locked ytterbium-doped single-mode fiber oscillator pumped by polarization-combined 915 nm laser diodes,” Opt. Lett. 37(19), 3972–3974 (2012). [CrossRef] [PubMed]

12.

M. E. Fermann and I. Hartl, “Ultrafast Fiber Laser Technology,” IEEE J. Sel. Top. Quantum Electron. 15(1), 191–206 (2009). [CrossRef]

13.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

14.

O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,” J. Lightwave Technol. 21(1), 61–68 (2003). [CrossRef]

15.

P. C. Chou, H. A. Haus, and J. F. Brennan III, “Reconfigurable time-domain spectral shaping of an optical pulse stretched by a fiber Bragg grating,” Opt. Lett. 25(8), 524–526 (2000). [CrossRef] [PubMed]

16.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]

17.

We remind that phase-sensitive and phase-insensitive configurations are commonly differentiated by looking at the input of the amplifier: in the first case, both the idler and the signal are launched with a specific phase relationship, while in the second case, only the signal is launched inside the amplifier.

18.

L. Grüner-Nielsen, D. Jakobsen, K. G. Jespersen, and B. Pálsdóttir, “A stretcher fiber for use in fs chirped pulse Yb amplifiers,” Opt. Express 18(4), 3768–3773 (2010). [CrossRef] [PubMed]

19.

J. Prawiharjo, N. K. Daga, R. Geng, J. H. Price, D. C. Hanna, D. J. Richardson, and D. P. Shepherd, “High fidelity femtosecond pulses from an ultrafast fiber laser system via adaptive amplitude and phase pre-shaping,” Opt. Express 16(19), 15074–15089 (2008). [CrossRef] [PubMed]

20.

D. N. Papadopoulos, I. Martial, M. Hanna, F. Druon, and P. Georges, “Active spectral phase control by use of an acousto-optic programmable filter in high-repetition-rate sub-80 fs nonlinear fiber amplifiers,” Opt. Lett. 33(13), 1431–1433 (2008). [CrossRef] [PubMed]

21.

J. van Howe, G. Zhu, and C. Xu, “Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems,” Opt. Lett. 31(11), 1756–1758 (2006). [CrossRef] [PubMed]

22.

J. W. Dawson, M. J. Messerly, H. H. Phan, J. K. Crane, R. J. Beach, C. W. Siders, and C. Barty, “High-Energy, Short-Pulse Fiber Injection Lasers at Lawrence Livermore National Laboratory,” IEEE J. Sel. Top. Quantum Electron. 15(1), 207–219 (2009). [CrossRef]

23.

Y. Zaouter, L. P. Ramirez, D. N. Papadopoulos, C. Hönninger, M. Hanna, F. Druon, E. Mottay, and P. Georges, “Temporal cleaning of a high-energy fiber-based ultrafast laser using cross-polarized wave generation,” Opt. Lett. 36(10), 1830–1832 (2011). [CrossRef] [PubMed]

24.

A. Mussot, A. Kudlinski, and E. Hugonnot, “Procédé and dispositif d’amplification paramétrique optique d’impulsions à dérive en fréquence, utilisant deux signaux de pompe and permettant l’élargissement de la bande spectrale de gain,” U.S. Patent FR 11 61642 (dec 14, 2011).

OCIS Codes
(140.3280) Lasers and laser optics : Laser amplifiers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 21, 2012
Revised Manuscript: January 23, 2013
Manuscript Accepted: January 24, 2013
Published: May 10, 2013

Citation
Arnaud Mussot, Alexandre Kudlinski, Patrick Beaure d’Augères, and Emmanuel Hugonnot, "Amplification of ultra-short optical pulses in a two-pump fiber optical parametric chirped pulse amplifier," Opt. Express 21, 12197-12203 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12197


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References

  1. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, 1st ed. (Cambridge University Press, 2007).
  2. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8(3), 506–520 (2002). [CrossRef]
  3. S. Radic, “Parametric Signal Processing,” IEEE J. Sel. Top. Quantum Electron.18(2), 670–680 (2012). [CrossRef]
  4. A. Vedadi, M. Jamshidifar, and M. E. Marhic, “Continuous-wave one-pump fiber optical parametric amplifier with 230 nm gain bandwidth,” paper 1.1.4 in 35th European Conference on Optical Communication (ECOC), p. 1–2 (2009).
  5. R. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron.11(3), 100–103 (1975). [CrossRef]
  6. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron.8(3), 538–547 (2002). [CrossRef]
  7. C. Caucheteur, D. Bigourd, E. Hugonnot, P. Szriftgiser, A. Kudlinski, M. Gonzalez-Herraez, and A. Mussot, “Experimental demonstration of optical parametric chirped pulse amplification in optical fiber,” Opt. Lett.35(11), 1786–1788 (2010). [CrossRef] [PubMed]
  8. D. Bigourd, L. Lago, A. Mussot, A. Kudlinski, J.-F. Gleyze, and E. Hugonnot, “High-gain fiber, optical-parametric, chirped-pulse amplification of femtosecond pulses at 1 μm,” Opt. Lett.35(20), 3480–3482 (2010). [CrossRef] [PubMed]
  9. D. Bigourd, L. Lago, A. Kudlinski, E. Hugonnot, and A. Mussot, “Dynamics of fiber optical parametric chirped pulse amplifiers,” J. Opt. Soc. Am. B28(11), 2848–2854 (2011). [CrossRef]
  10. 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). [CrossRef] [PubMed]
  11. T. Kurita, H. Yoshida, T. Kawashima, and N. Miyanaga, “Generation of sub-7-cycle optical pulses from a mode-locked ytterbium-doped single-mode fiber oscillator pumped by polarization-combined 915 nm laser diodes,” Opt. Lett.37(19), 3972–3974 (2012). [CrossRef] [PubMed]
  12. M. E. Fermann and I. Hartl, “Ultrafast Fiber Laser Technology,” IEEE J. Sel. Top. Quantum Electron.15(1), 191–206 (2009). [CrossRef]
  13. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).
  14. O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,” J. Lightwave Technol.21(1), 61–68 (2003). [CrossRef]
  15. P. C. Chou, H. A. Haus, and J. F. Brennan, “Reconfigurable time-domain spectral shaping of an optical pulse stretched by a fiber Bragg grating,” Opt. Lett.25(8), 524–526 (2000). [CrossRef] [PubMed]
  16. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B27(11), B63–B92 (2010). [CrossRef]
  17. We remind that phase-sensitive and phase-insensitive configurations are commonly differentiated by looking at the input of the amplifier: in the first case, both the idler and the signal are launched with a specific phase relationship, while in the second case, only the signal is launched inside the amplifier.
  18. L. Grüner-Nielsen, D. Jakobsen, K. G. Jespersen, and B. Pálsdóttir, “A stretcher fiber for use in fs chirped pulse Yb amplifiers,” Opt. Express18(4), 3768–3773 (2010). [CrossRef] [PubMed]
  19. J. Prawiharjo, N. K. Daga, R. Geng, J. H. Price, D. C. Hanna, D. J. Richardson, and D. P. Shepherd, “High fidelity femtosecond pulses from an ultrafast fiber laser system via adaptive amplitude and phase pre-shaping,” Opt. Express16(19), 15074–15089 (2008). [CrossRef] [PubMed]
  20. D. N. Papadopoulos, I. Martial, M. Hanna, F. Druon, and P. Georges, “Active spectral phase control by use of an acousto-optic programmable filter in high-repetition-rate sub-80 fs nonlinear fiber amplifiers,” Opt. Lett.33(13), 1431–1433 (2008). [CrossRef] [PubMed]
  21. J. van Howe, G. Zhu, and C. Xu, “Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems,” Opt. Lett.31(11), 1756–1758 (2006). [CrossRef] [PubMed]
  22. J. W. Dawson, M. J. Messerly, H. H. Phan, J. K. Crane, R. J. Beach, C. W. Siders, and C. Barty, “High-Energy, Short-Pulse Fiber Injection Lasers at Lawrence Livermore National Laboratory,” IEEE J. Sel. Top. Quantum Electron.15(1), 207–219 (2009). [CrossRef]
  23. Y. Zaouter, L. P. Ramirez, D. N. Papadopoulos, C. Hönninger, M. Hanna, F. Druon, E. Mottay, and P. Georges, “Temporal cleaning of a high-energy fiber-based ultrafast laser using cross-polarized wave generation,” Opt. Lett.36(10), 1830–1832 (2011). [CrossRef] [PubMed]
  24. A. Mussot, A. Kudlinski, and E. Hugonnot, “Procédé and dispositif d’amplification paramétrique optique d’impulsions à dérive en fréquence, utilisant deux signaux de pompe and permettant l’élargissement de la bande spectrale de gain,” U.S. Patent FR 11 61642 (dec 14, 2011).

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