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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 15074–15089
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High fidelity femtosecond pulses from an ultrafast fiber laser system via adaptive amplitude and phase pre-shaping

Jerry Prawiharjo, Nikita K. Daga, Rui Geng, Jonathan H.V. Price, David C. Hanna, David J. Richardson, and David P. Shepherd  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 15074-15089 (2008)
http://dx.doi.org/10.1364/OE.16.015074


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Abstract

The generation of high-fidelity femtosecond pulses is experimentally demonstrated in a fiber based chirped-pulse amplification (CPA) system through an adaptive amplitude and phase pre-shaping technique. A pulse shaper, based on a dual-layer liquid crystal spatial light modulator (LC-SLM), was implemented in the fiber CPA system for amplitude and phase shaping prior to amplification. The LC-SLM was controlled using a differential evolution algorithm, to maximize a two-photon absorption detector signal from the compressed fiber CPA output pulses. It is shown that this approach compensates for both accumulated phase from material dispersion and nonlinear phase modulation. A train of pulses was produced with an average power of 12.6W at a 50MHz repetition rate from our fiber CPA system, which were compressible to high fidelity pulses with a duration of 170 fs.

© 2008 Optical Society of America

1. Introduction

Rare-earth doped fiber laser systems are a promising alternative to bulk ultrafast laser systems [1

1. J. Limpert, F. Röser, T. Schreiber, and A. Tünnermann, “High-power ultrafast fiber laser systems,” IEEE J. of Sel. Top. in Quantum Electron. 12, 233–244 (2006). [CrossRef]

], whose power scaling is not straightforward due to their generally low single-pass gain, coupled with thermo-optical issues. Ytterbium-doped fibers are particularly interesting due to their broad emission spectrum, allowing the generation and amplification of ultrashort optical pulses, although the gain medium does not support the generation of pulses as short as can be obtained with Ti:sapphire. The fiber geometry offers good thermo-optical properties, a high single-pass gain, excellent output beam quality, and coupled with continuous-wave diode pumping, has allowed the realization of compact ultrafast laser systems with more than 100W average power at various repetition rates [2

2. F. Röser, J. Rothhard, B. Ortaç, A. Liem, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “131W 220 fs fiber laser system,” Opt. Lett. 30, 2754–2756 (2005). [CrossRef] [PubMed]

, 3

3. F. Röser, D. Schimpf, O. Schmidt, B. Ortaç, K. Rademaker, J. Limpert, and A. Tünnermann, “90W average power 100 µJ energy femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32, 2230–2232 (2007). [CrossRef] [PubMed]

]. Energy levels reaching the millijoule regime have also recently been demonstrated, thanks to the utilization of novel fiber designs, such as photonic crystal-fiber (PCF) [4

4. F. Röser, T. Eidam, J. Rothhard, O. Schmidt, D. N. Schimpf, J. Limpert, and A. Tünnermann, “Millijoule pulse energy high repetition rate femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32, 3495–3497 (2007). [CrossRef] [PubMed]

].

The chirped-pulse amplification (CPA) technique is the preferred way to achieve high peak-power ultrashort pulses in ultrafast laser systems [5

5. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

]. However, a multitude of factors can degrade the pulse quality, such as uncompensated material dispersion, nonlinearity, and a non-uniform spectral gain profile with finite width. The pulse quality degradation is most notably manifested in the presence of a pedestal, which limits the maximum peak intensity of the pulses that can be achieved [6

6. Y. H. Chuang, D. D. Meyerhofer, S. Augst, H. Chen, J. Peatross, and S. Uchida, “Supression of the pedestal in a chirped-pulse-amplification laser,” J. Opt. Soc. Am. B 8, 1226–1235 (1991). [CrossRef]

]. Two factors leading to pulse degradation are uncompensated higher-order spectral phase between stretcher, compressor, and amplifier materials, and self-phase-modulation (SPM), which leads to a nonlinear spectral phase [7

7. M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. 19, 2149–2151 (1994). [CrossRef] [PubMed]

].

In ultrafast bulk solid-state laser systems, a great deal of effort has been spent to design appropriate stretcher and compressor pairs to minimize the uncompensated higher-order spectral phase [8

8. B. E. Lemoff and C. P. J. Barty, “Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses,” Opt. Lett. 18, 1651–1653 (1993). [CrossRef] [PubMed]

]. However, this approach requires careful characterization and design of the system. Furthermore, it is non-adjustable and thus prevents easy reconfiguration of the system. Ultimately, programmable femtosecond pulse shaping [9

9. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulator,” Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]

] offers the possibility of almost arbitrary modifications of the phase and amplitude of ultrashort optical pulses, and thus eliminate the need for meticulous characterization and design of the entire system. In particular, in combination with adaptive learning loop utilizing optimization algorithms, i.e. adaptive pulse shaping, it has allowed for the generation of high-fidelity pulses [10

10. D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. 22, 1793–1795 (1997). [CrossRef]

, 11

11. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65, 779–782 (1997). [CrossRef]

, 12

12. T. Brixner, M. Strehle, and G. Gerber, “Feedback-controlled optimization of amplified femtosecond laser pulses,” Appl. Phys. B 68, 281–284 (1999). [CrossRef]

, 13

13. K. H. Hong and C. H. Nam, “Adaptive pulse compression of femtosecond laser pulse using a low-loss pulse shaper,” Jpn. J. Appl. Phys. 43, 5289–5293 (2004). [CrossRef]

, 14

14. R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74, 2670–2674 (2003). [CrossRef]

]. This approach also allows for the generation of arbitrary pulse shapes, using either phase-only modulation [15

15. T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000). [CrossRef]

], or phase and amplitude modulation [16

16. K. Ohno, T. Tanabe, and F. Kannari, “Adaptive pulse shaping of phase and amplitude of an amplified femtosecond pulse laser by direct reference to frequency-resolved optical gating traces,” J. Opt. Soc. Am. B 19, 2781–2790 (2002). [CrossRef]

]. Nevertheless, femtosecond pulse shaping typically suffers from low throughput, associated with various losses in the setup, and usually a low damage threshold from the programmable modulator, e.g. a pixelated liquid crystal array. In order to circumvent this problem, the pulse shaper has to be incorporated into the system before any high power amplification stages. In Ti:sapphire laser systems, a phase-only pulse shaping setup has been successfully incorporated as part of the stretcher with [17

17. A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, “Adaptive control of pulse phase in a chirped-pulse amplifier,” Opt. Lett. 23, 1915–1917 (1998). [CrossRef]

, 18

18. A. Efimov and D. H. Reitze, “Programmable dispersion compensation and pulse shaping in a 26-fs chirped-pulse amplifier,” Opt. Lett. 23, 1612–1614 (1998). [CrossRef]

, 19

19. A. Efimov, M. D. Moores, B. Mei, J. L. Krause, C. W. Siders, and D. H. Reitze, “Minimization of dispersion in an ultrafast chirped pulse amplifier using adaptive learning,” Appl. Phys. B 70, S133–S141 (2000). [CrossRef]

], or without [20

20. G. Chériaux, O. Albert, V. Wänman, J. P. Chambaret, C. Félix, and G. Mourou, “Temporal control of amplified femtosecond pulses with a deformable mirror in a stretcher,” Opt. Lett. 26, 169–171 (2001). [CrossRef]

], global optimization algorithms to minimize the pulse duration at the output. The generation of arbitrary shaped pulses has also been demonstrated using amplitude and phase shaping and a global optimization algorithm [21

21. T. Tanabe, K. Ohno, T. Okamoto, M. Yamanaka, and F. Kannari, “Feedback control for accurate shaping of ultrashort optical pulses prior to chirped pulse amplification,” Japanese Journal of Applied Physics 43, 1366–1375 (2004). [CrossRef]

].

In ultrafast fiber laser systems, material dispersion, nonlinearity, and a non-uniform spectral gain profile with finite width, are critical considerations, because of the optical confinement and long interaction length in the fiber geometry. As temporal stretching is physically limited by the finite size of the grating compressor, while the scaling of large-mode-area (LMA) fibers will eventually undermine the advantages of fiber geometry, most of the work has concentrated at managing the SPM, by compensation of the nonlinear phase induced by SPM via the third-order material dispersion [22

22. S. Zhou, L. Kuznetsova, A. Chong, and F. W. Wise, “Compensation of nonlinear phase shifts with third-order dispersion in short-pulse fiber amplifiers,” Opt. Express 13, 4869–4877 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-4869 [CrossRef] [PubMed]

, 23

23. L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, “High energy femtosecond Yb cubicon fiber amplifier,” Opt. Express 13, 4717–4722 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4717 [CrossRef] [PubMed]

, 24

24. A. Chong, L. Kuznetsova, and F. W. Wise, “Theoretical optimization of nonlinear chirped-pulse fiber amplifiers,” J. Opt. Soc. Am. B 24, 1815–1823 (2007). [CrossRef]

, 25

25. L. Kuznetsova and F. W. Wise, “Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification,” Opt. Lett. 32, 2671–2673 (2007). [CrossRef] [PubMed]

], or utilization of the interplay between the SPM-induced spectral broadening and gain shaping [26

26. L. Kuznetsova, A. Chong, and F. W. Wise, “Interplay of nonlinearity and gain shaping in femtosecond fiber amplifiers,” Opt. Lett. 31, 2640–2642 (2006). [CrossRef] [PubMed]

, 27

27. T. Schreiber, D. Schimpf, D. Müller, F. Röser, J. Limpert, and A. Tünnermann, “Influence of pulse shape in self-phase-modulation-limited chirped pulse fiber amplifier systems,” Journal of the Optical Society of America B 24, 1809–1814 (2007). [CrossRef]

]. These approaches, however, cannot fully compensate the nonlinear phase-modulation due to the SPM, and require careful design of the laser system. Recently, another method was proposed to actively compensate for the SPM using phase modulation of the stretched pulses [28

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

, 29

29. G. Zhu, J. Edinberg, and C. Xu, “Nonlinear distortion free fiber-based chirped pulse amplification with self-phase modulation up to 2π,” Opt. Express 15, 2530–2534 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2530 [CrossRef] [PubMed]

], but the phase modulator can only impose a limited amount of phase shift and is limited in terms of complexity of phase profile that can be applied. Finally, the effects of the non-uniform spectral gain profile with finite width also becomes more prominent, further degrading the pulse quality. In fiber CPA systems, multiple amplifier stages are usually necessary to achieve the desired power level. Hence, fiber CPA systems present more technical challenges in producing high-fidelity pulses.

In view of the work in ultrafast bulk solid-state laser systems, adaptive pulse shaping prior to amplification has been adopted in fiber CPA laser systems with some success. Recently, amplitude-only shaping has been demonstrated to control the nonlinear-phase modulation induced by SPM at low energy [30

30. D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express 15, 16 945–16 953 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16945

], but it cannot compensate for higher-order spectral phase due to the material dispersion. Our group recently demonstrated a phase-only shaping in a high energy fiber laser system [31

31. F. He, H. S. S. Hung, J. H. V. Price, N. K. Daga, N. Naz, J. Prawiharjo, D. C. Hanna, D. P. Shepherd, D. J. Richardson, J. W. Dawson, C. W. Siders, and C. P. J. Barty, “High energy femtosecond fiber chirped pulse amplification system with adaptive phase control,” Opt. Express (2008). http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5813 [CrossRef] [PubMed]

].

In this paper, we demonstrate, for the first time to our knowledge, the implementation of adaptive amplitude and phase pre-shaping in a fiber-based CPA system to generate high-fidelity compressed femtosecond pulses. A pulse shaper based on a dual-layer liquid crystal spatial light modulator (LC-SLM) was implemented in our fiber CPA system for amplitude and phase shaping prior to amplification. The LC-SLM is compatible with pulses at any repetition rate, making it the preferred choice for our experiments, and contrasts with a recent post-amplification shaping work utilizing dazzler with limited repetition rate and output efficiency [32

32. 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, 1431–1433 (2008). [CrossRef] [PubMed]

]. The LC-SLM was controlled using a differential evolution (DE) algorithm, to maximize a two-photon absorption detector signal produced from the compressed fiber CPA output pulses. We show that our approach compensates for both accumulated phase from higher-order material dispersion and nonlinear phase modulation. A train of pulses with an average power of 12.6W at a 50MHz repetition rate was produced from the fiber CPA system, compressible to high fidelity pulses with a 170 fs temporal full-width at half-maximum (FWHM).

This paper is organized as follows. The fiber CPA experimental setup and the implementation of the adaptive loop pulse shaping are described in Section 2. The experimental results are presented and discussed in Section 3. Finally, we conclude the paper in Section 4.

2. Experimental Setup

2.1. Ultrafast fiber laser system

Fig. 1. Schematic illustration of the ultrafast fiber laser system. HWP: Half-wave plate, QWP: Quarter-wave plate, OI: Optical isolator, WDM: Wavelength division multiplexer, SM: Single-mode, PBS: Polarizing beam splitter, FR: Faraday rotator, APP: Anamorphic prism pair, G: Gratings, CM: Cylindrical mirror, SLM: Spatial light modulator, PCF: Photonic crystal fiber, T: Telescope arrangement, TPA: Two-photon absorption detector, SHG FROG: Second-harmonic generation frequency-resolved optical gating, DE: Differential evolution algorithm on a computer.

2.2. Adaptive loop

Fig. 2. Illustration of the applied voltage on the two layers of the LC-SLM, M0 and M1, as interpolated from the pixels controlled by the adaptive algorithm, indicated by the open circles.

We argue that this approach has advantages over parametrization with a truncated Taylor series. Firstly, in terms of optimization, changing the value of a parameter in a Taylor series would completely change the entire profile, while in the case of interpolation, the change would be local. Secondly, in practice, the continuous profile inferred from the Taylor series does not necessarily correspond to the one applied to the SLM, due to the SLM spatial pixellization and driving voltage discretization. Finally, while it is easy to individually calculate the boundaries that must be placed on each Taylor series coefficient due to the physical limitations imparted by the pixellization of the SLM, it is not easy to calculate such limits when many coefficients need to be applied simultaneously.

The problem was formulated as max{f(X)|X}, where X is an integer vector of 1 × 2N c parameters, bounded between 600 and 4096, i.e. each of its component X j∈ [600,4096], where j = 1,…,2N c, and f(X) is the evaluated TPA detector signal from the applied voltage. Note that X is a concatenation of the N c pixel voltages from the two layers of the LC-SLM. The DE algorithm [36

36. R. Storn and K. Price, “Differential evolution - A simple and efficient heuristic for global optimization over continuous space,” J. Glob. Optim. 11, 341–359 (1997). [CrossRef]

] was implemented as our global optimization algorithm in Matlab, which was also used to control the SLM, and to read the TPA detector signal. The DE algorithm was chosen, because it has been shown to consistently outperform simulated annealing or genetic algorithms in most cases [36

36. R. Storn and K. Price, “Differential evolution - A simple and efficient heuristic for global optimization over continuous space,” J. Glob. Optim. 11, 341–359 (1997). [CrossRef]

, 37

37. M. A. Ali, C. Khompatraporn, and Z. B. Zabinsky, “A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems,” J. Glob. Optim. 31, 635–672 (2005). [CrossRef]

], which we have confirmed both in many simulations and experiments.

Vi,g+1={Xr1,g+[F+(1F)ug](Xr2,gXr3,g)ifug>0.04,(Xr1,g+Xr2,g+Xr3,g)3+(p1p2)(Xr1,gXr2,g)+(p2p3)(Xr2,gXr3,g)+(p3p1)(Xr3,gXr1,g)otherwise,
(1)

where

p1=f(Xr1,g)/p,p2=f(Xr2,g)/p,p3=f(Xr3,g)/p,
(2)

and

p=f(Xr1,g)+f(Xr2,g)+f(Xr3,g)
(3)

In the above equations, u g,u′ g∈ [0,1

1. J. Limpert, F. Röser, T. Schreiber, and A. Tünnermann, “High-power ultrafast fiber laser systems,” IEEE J. of Sel. Top. in Quantum Electron. 12, 233–244 (2006). [CrossRef]

] are uniformly distributed real random numbers, drawn every generation, and F is called the scaling factor, which is a control parameter of the DE algorithm.

Wj,i,g+1={Vj,i,g+1ifuCrj=w,Xj,i,gotherwise,
(4)

Xi,g+1={Wi,g+1iff(Wi,g+1)f(Xi,g),Xi,gotherwise.
(5)

In all of our experiments, we chose F = 0.75, C r = 0.5, and N p = 30. In our experiments, the optimization took an average of 0.145 minutes per generation, which was mostly spent updating the SLM, integrating the detector signal, and on electronic communication with the instruments. Therefore, we did not implement a stopping condition based on convergence criteria, but, instead, we ran the DE algorithm for a specific number of generations. In each optimization, we monitored the diversity among the individuals in the population to check for the convergence.

3. Results and discussions

Fig. 3. Normalized measured spectra at the output of the oscillator, after the first preamplifier and the pulse shaper, and after the second pre-amplifier in logarithmic (a) and linear (b) scale.

Figure 3 shows typical measured spectra at the output of the oscillator, after the first preamplifier and the pulse shaper, and after the second pre-amplifier. It is worth noting, at this point, that the fringes that appear on the oscillator spectrum, are possibly due to the etalon effect. The fringes grow in subsequent amplifier stages, owing to the SPM [39]. These fringes could not be eliminated experimentally. The effect of the non-uniform spectral gain profile with finite width is evident, as the spectral FWHM of the pulse was reduced from 16nm at the output of oscillator, to 12 nm after the first pre-amplifier, and to 11nm after the second-preamplifier. Nevertheless, the 20dB spectral width of 23nm was maintained.

Firstly, the pulses from the compressor were characterized without intentional shaping by the pulse shaper (by not applying any voltage to the SLM). The final amplifier was pumped to produce a train of pulses with an average power of 2.3W prior to the compressor, and then the separation of the grating pair in the compressor was adjusted to maximize the intensity of the TPA detector, and the second harmonic signal from the SHG FROG setup when the pulses from both arms were temporally overlapped. This resulted in a grating separation of 10.2 cm. The output pulses were then characterized at two power levels using the SHG FROG;

at 2.3W and at 12.6W average power, without changing the grating separation. We shall refer to these as the low and high average power, respectively, throughout the remainder of this paper. Figure 4(a) and (b) show the square-root of the measured SHG FROG traces, after interpolation onto a 128 × 128 Fourier grid, for both cases. Plotting the square root of the FROG trace was aimed at emphasizing the detail at low intensity. The spectral and the temporal intensity were then retrieved, as well as the group delay [dϕ (ω)/dω] and the instantaneous frequency [-1/(2π)×dϕ(t)/dt], from these traces, as shown in Fig. 4(c) to (f). The root-mean-square (rms) retrieval errors of these traces were less than 8 × 10-3. The excellent agreement between the retrieved and measured spectra, in Fig. 4(c) and (d), demonstrates the quality of the home-built SHG FROG.

Fig. 4. (a,b) Contour plot of square-root of measured SHG FROG traces, after interpolation onto a 128 × 128 Fourier grid, of the pulses after the compressor without intentional shaping. The contour lines represent levels [0.02,0.06,0.1,0.2,…,1]. (c,d) Retrieved spectral intensity (blue curves), spectral group delay (green curves), and measured spectra (red curves). (e,f) Retrieved temporal intensity (blue curves) and instantaneous frequency (green curves). The figures correspond to measured average powers of 2.3W (a,c,e) and 12.6W (b,d,f) prior to the compressor.

The mainly parabolic profile of group delays shown in Fig. 4(c) and (d), and the side-lobes in the temporal intensity profiles shown in Fig. 4(e) and (f), are a strong indication that the dominant effect on the output pulses of the CPA system was the accumulated third-order dispersion (TOD). These results are expected, since there was no compensation element for the TOD placed in our system. In addition, the departure of the group delay profiles from parabolic at both low and high average power levels [Fig. 4(c,d)] suggests the presence of nonlinear phase-modulation, whose amount increases with power level, as signified by the spectral broadening that accompanied the increase in the average power. This increase in accumulated nonlinear phase modulation causes the side-lobes of the temporal profile in the high power case to not decrease monotonically, as seen in Fig. 4(f).

The accumulated nonlinear-phase modulation evident in the above characterization is expected, since there was no specific stretching of the pulses prior to amplification in our fiber CPA system. The estimated upper limit of the accumulated nonlinear phase was ϕNL=π rad at low power, and ϕNL = 1.6π rad at high power, of which the contribution prior to the pulse shaper was 0.7π rad. In this calculation, an exponential amplification with a constant gain per unit length with flat spectral gain profile in the fiber amplifiers was assumed. The accumulated nonlinear phase, coupled with non-uniform spectral gain profile, caused the large change in spectral intensity after the different amplifier stages. It is worth noting that we observed nonlinear polarization rotation (NPR) in both of our pre-amplifiers. Due to the NPR, environmental changes, particularly of temperature, induce small fluctuations in the output of our system, explaining the discrepancy between the measured spectra after the pulse shaper and the second pre-amplifier in Fig. 7 and 10. However, this fluctuation is a very slow process that happens on a day-to-day timescale, and thus did not affect our experiments.

Figure 5 shows the evolution of the TPA detector signal, normalized to the case without intentional shaping, evaluated from the applied SLM voltages of the best individual in the population at each generation. Note that since the algorithm was run from a random initial condition, the best individuals at the early stages of the optimization had lower TPA detector signal than the case without intentional shaping. After 350 generations, there was not much diversity among individuals in the population, indicating that the algorithm had converged to a solution, and the TPA signal was improved by a factor of 4.2. The mask corresponding to the best individual in the population was then applied to the SLM, and the pulses were then characterized using the SHG FROG. Figure 6(a) shows square-root of the measured FROG trace, after interpolation onto a 128 × 128 Fourier grid. The retrieved spectral intensity and group delay are shown in Fig. 6(b), while the temporal intensity and instantaneous frequency are shown in Fig. 6(c). The rms retrieval error was less than 2 × 10-3, and the temporal FWHM of the retrieved intensity is 195 fs. The calculated Fourier transform-limited profile is also shown in Fig. 6(c). The excellent agreement between the retrieved and calculated transform-limited profiles down to the -20 dB level, highlights the success of the optimization algorithm. The measured spectra at various points in our system are shown in Fig. 7, before and after optimization, as well as the calculated transmission and group delay applied by the SLM. The optimization yielded a spectral broadening toward shorter wavelengths after the second pre-amplifier, as well as in the final spectrum.

Fig. 5. Two-photon absorption detector signal, normalized to the case of without intentional shaping, and evaluated from the best individual in the population as a function of generation, for average power of 2.3W (blue dots) and 12.6W (red dots). For the low average power case, a constant number of controlled pixels, N c = 15, was used throughout the optimization. For the high average power case, the number of controlled pixels N c were increased from 15 to 51 during the optimization, at positions indicated by the arrows (see text).
Fig. 6. (a) Contour plot of square-root of measured SHG FROG trace, after interpolation onto a 128 × 128 Fourier grid, of the compressed low average power pulses, after the maximization of the TPA signal by controlling every 8th pixels of the SLM (see text). The contour lines represent levels [0.02,0.06,0.1,0.2,…,1]. (b) Retrieved spectral intensity (blue curve), spectral group delay (green curve), and measured spectrum (red curve). (c) Retrieved temporal intensity (blue curve) and instantaneous frequency (green curve), as well as the calculated transform-limited intensity profile (red curve).
Fig. 7. (a) Calculated applied group delay (blue curve) for the low average power pulses overlayed with the negative of the retrieved group delay for the case without intentional shaping [Fig. 4(c)]. The measured shaped spectrum after the shaper (shaded grey) is shown for reference. (b) Calculated transmission (black curve) of the SLM after maximizing the TPA detector signal. Normalized measured spectra before (shaded) and after (curves) optimization after the pulse shaper, after the second pre-amplifier, and after the compressor are shown, as indicated by the labels.

Fig. 8. Same as Fig. 6, but for high average power, after the maximization of the TPA signal by controlling every 8th pixels of the SLM (see text). Note that the temporal intensity in (c) is shown in linear scale.

Finer control over the pulse shaping was then attempted by increasing the number of controlled pixels, from every 8th pixel, to every 4th pixel at generation 51, and finally to every 2nd pixel at generation 101, between pixels 16 to 112 of the SLM. In addition, pixels number 12 and 116 were still controlled, resulting in a final total of N c = 51 controlled pixels on each layer of the SLM. The DE algorithm was run with this condition for 450 generations, taking 65 minutes to complete. The evolution of the TPA signal evaluated from the best individual in the population, normalized to the case without intentional shaping, at each generation is shown in Fig. 5 (red dots).

Fig. 9. Same as Fig. 6, but for high average power, after the maximization of the TPA signal with increasing numbers controlled pixels of the SLM (see text).

Although the DE algorithm started with random initial candidate solutions, the optimization results had a high reproducibility. The resulting applied phase and transmission profiles showed little dependence on the initial condition, yielding similar compressed pulse profiles in each case, which implies that our results have to be in the vicinity of the global optimum. In fact, there is a consistency in the optimized spectra after the shaper in both low and high power cases, as shown in Fig. 7 and 10, i.e. suppression of the part of the pulse spectra between 1035 and 1040 nm. In order to obtain the true global optimum, more generations would be required, but with diminishing returns that may not justify the effort. It is important to note that it is not necessary to start from random initial candidate solutions. It is possible to feed a previously optimized data as one of the initial candidate solutions in order to reduce the optimization time.

Fig. 10. Same as Fig. 7, but for the case of high average power, after the maximization of the TPA signal with increasing controlled pixels of the SLM [Fig. 9].

In order to achieve high fidelity pulses, maximization of a TPA detector signal or, similarly, a SHG signal, is the simplest objective function to be implemented. Note that the maximization of TPA signal does not guarantee the generation of a specific spectral shape at the output. The approach in Ref. [30

30. D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express 15, 16 945–16 953 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16945

] to produce a parabolic shaped spectrum after nonlinear amplification via an amplitude-only pre-shaping in order to obtain high fidelity pulses after compression can work to a certain extent. However, as the authors themselves have pointed out, this method requires more effort to compensate for accumulated higher-order phase from the material dispersion. Furthermore, when a more complex setup is involved, comprising multiple amplifier stages such as in our case, this approach is less likely to be successful.

4. Conclusion

In conclusion, we have successfully demonstrated an adaptive amplitude and phase pre-shaping technique for producing high-fidelity femtosecond pulses in a fiber CPA system. We have demonstrated that this technique is very robust, effective, and efficient, in compensating for both accumulated phase from higher-order material dispersion and nonlinear phase-modulation. We did not have to perform painstaking characterization and design to optimize our fiber CPA system to produce the high-fidelity femtosecond pulses presented here. We found that, with increasing nonlinearity, a finer control over the pulse shaping was necessary to achieve high fidelity pulses upon compression. This technique should enable power-scaling to higher energy and/or average power at various repetition rates. In addition to producing high-fidelity compressed pulses, this technique has the potential to produce arbitrary shaped pulses necessary in various applications, including coherent control [40

40. B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63, 063412/1–12 (2001). [CrossRef]

].

Acknowledgment

This work was supported by EPSRC Instrument Grant EP/C009479/1. J.H.V. Price is supported by a Royal Academy of Engineering/EPSRC research fellowship.

References and links

1.

J. Limpert, F. Röser, T. Schreiber, and A. Tünnermann, “High-power ultrafast fiber laser systems,” IEEE J. of Sel. Top. in Quantum Electron. 12, 233–244 (2006). [CrossRef]

2.

F. Röser, J. Rothhard, B. Ortaç, A. Liem, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “131W 220 fs fiber laser system,” Opt. Lett. 30, 2754–2756 (2005). [CrossRef] [PubMed]

3.

F. Röser, D. Schimpf, O. Schmidt, B. Ortaç, K. Rademaker, J. Limpert, and A. Tünnermann, “90W average power 100 µJ energy femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32, 2230–2232 (2007). [CrossRef] [PubMed]

4.

F. Röser, T. Eidam, J. Rothhard, O. Schmidt, D. N. Schimpf, J. Limpert, and A. Tünnermann, “Millijoule pulse energy high repetition rate femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32, 3495–3497 (2007). [CrossRef] [PubMed]

5.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

6.

Y. H. Chuang, D. D. Meyerhofer, S. Augst, H. Chen, J. Peatross, and S. Uchida, “Supression of the pedestal in a chirped-pulse-amplification laser,” J. Opt. Soc. Am. B 8, 1226–1235 (1991). [CrossRef]

7.

M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. 19, 2149–2151 (1994). [CrossRef] [PubMed]

8.

B. E. Lemoff and C. P. J. Barty, “Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses,” Opt. Lett. 18, 1651–1653 (1993). [CrossRef] [PubMed]

9.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulator,” Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]

10.

D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. 22, 1793–1795 (1997). [CrossRef]

11.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65, 779–782 (1997). [CrossRef]

12.

T. Brixner, M. Strehle, and G. Gerber, “Feedback-controlled optimization of amplified femtosecond laser pulses,” Appl. Phys. B 68, 281–284 (1999). [CrossRef]

13.

K. H. Hong and C. H. Nam, “Adaptive pulse compression of femtosecond laser pulse using a low-loss pulse shaper,” Jpn. J. Appl. Phys. 43, 5289–5293 (2004). [CrossRef]

14.

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74, 2670–2674 (2003). [CrossRef]

15.

T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000). [CrossRef]

16.

K. Ohno, T. Tanabe, and F. Kannari, “Adaptive pulse shaping of phase and amplitude of an amplified femtosecond pulse laser by direct reference to frequency-resolved optical gating traces,” J. Opt. Soc. Am. B 19, 2781–2790 (2002). [CrossRef]

17.

A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, “Adaptive control of pulse phase in a chirped-pulse amplifier,” Opt. Lett. 23, 1915–1917 (1998). [CrossRef]

18.

A. Efimov and D. H. Reitze, “Programmable dispersion compensation and pulse shaping in a 26-fs chirped-pulse amplifier,” Opt. Lett. 23, 1612–1614 (1998). [CrossRef]

19.

A. Efimov, M. D. Moores, B. Mei, J. L. Krause, C. W. Siders, and D. H. Reitze, “Minimization of dispersion in an ultrafast chirped pulse amplifier using adaptive learning,” Appl. Phys. B 70, S133–S141 (2000). [CrossRef]

20.

G. Chériaux, O. Albert, V. Wänman, J. P. Chambaret, C. Félix, and G. Mourou, “Temporal control of amplified femtosecond pulses with a deformable mirror in a stretcher,” Opt. Lett. 26, 169–171 (2001). [CrossRef]

21.

T. Tanabe, K. Ohno, T. Okamoto, M. Yamanaka, and F. Kannari, “Feedback control for accurate shaping of ultrashort optical pulses prior to chirped pulse amplification,” Japanese Journal of Applied Physics 43, 1366–1375 (2004). [CrossRef]

22.

S. Zhou, L. Kuznetsova, A. Chong, and F. W. Wise, “Compensation of nonlinear phase shifts with third-order dispersion in short-pulse fiber amplifiers,” Opt. Express 13, 4869–4877 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-4869 [CrossRef] [PubMed]

23.

L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, “High energy femtosecond Yb cubicon fiber amplifier,” Opt. Express 13, 4717–4722 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4717 [CrossRef] [PubMed]

24.

A. Chong, L. Kuznetsova, and F. W. Wise, “Theoretical optimization of nonlinear chirped-pulse fiber amplifiers,” J. Opt. Soc. Am. B 24, 1815–1823 (2007). [CrossRef]

25.

L. Kuznetsova and F. W. Wise, “Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification,” Opt. Lett. 32, 2671–2673 (2007). [CrossRef] [PubMed]

26.

L. Kuznetsova, A. Chong, and F. W. Wise, “Interplay of nonlinearity and gain shaping in femtosecond fiber amplifiers,” Opt. Lett. 31, 2640–2642 (2006). [CrossRef] [PubMed]

27.

T. Schreiber, D. Schimpf, D. Müller, F. Röser, J. Limpert, and A. Tünnermann, “Influence of pulse shape in self-phase-modulation-limited chirped pulse fiber amplifier systems,” Journal of the Optical Society of America B 24, 1809–1814 (2007). [CrossRef]

28.

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

29.

G. Zhu, J. Edinberg, and C. Xu, “Nonlinear distortion free fiber-based chirped pulse amplification with self-phase modulation up to 2π,” Opt. Express 15, 2530–2534 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2530 [CrossRef] [PubMed]

30.

D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express 15, 16 945–16 953 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16945

31.

F. He, H. S. S. Hung, J. H. V. Price, N. K. Daga, N. Naz, J. Prawiharjo, D. C. Hanna, D. P. Shepherd, D. J. Richardson, J. W. Dawson, C. W. Siders, and C. P. J. Barty, “High energy femtosecond fiber chirped pulse amplification system with adaptive phase control,” Opt. Express (2008). http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5813 [CrossRef] [PubMed]

32.

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, 1431–1433 (2008). [CrossRef] [PubMed]

33.

L. Lefort, J. H. V. Price, D. J. Richardson, G. J. Spühler, R. Paschotta, U. Keller, J. Fry, and A.R. Weston, “Practical low-noise stretched-pulse Yb3+-doped fiber laser,” Opt. Lett. 27, 291–293 (2002). [CrossRef]

34.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 3902–3905 (2004). [CrossRef]

35.

A. Präkelt, M. Wollenhaupt, A. Assion, C. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, “Compact, robust, and flexible setup for femtosecond pulse shaping,” Rev. Sci. Instrum. 74, 4950–4953 (2003). [CrossRef]

36.

R. Storn and K. Price, “Differential evolution - A simple and efficient heuristic for global optimization over continuous space,” J. Glob. Optim. 11, 341–359 (1997). [CrossRef]

37.

M. A. Ali, C. Khompatraporn, and Z. B. Zabinsky, “A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems,” J. Glob. Optim. 31, 635–672 (2005). [CrossRef]

38.

H. Y. Fan and J. Lampinen, “A trigonometric mutation operation to differential evolution,” J. Glob. Optim. 27, 105–129 (2003). [CrossRef]

39.

D. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Decrease of pulse-contrast in nonlinear chirped-pulse amplification systems due to high-frequency spectral phase ripple,” Opt. Express 16, 8876–8886 (2008). http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-8876 [CrossRef] [PubMed]

40.

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63, 063412/1–12 (2001). [CrossRef]

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(320.5540) Ultrafast optics : Pulse shaping
(320.7090) Ultrafast optics : Ultrafast lasers

ToC Category:
Ultrafast Optics

History
Original Manuscript: July 1, 2008
Revised Manuscript: August 15, 2008
Manuscript Accepted: August 15, 2008
Published: September 10, 2008

Citation
Jerry Prawiharjo, Nikita K. Daga, Rui Geng, Jonathan H. Price, David C. Hanna, David J. Richardson, and David P. Shepherd, "High fidelity femtosecond pulses from an ultrafast fiber laser system via adaptive amplitude and phase pre-shaping," Opt. Express 16, 15074-15089 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15074


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References

  1. J. Limpert, F. Röser, T. Schreiber, and A. Tünnermann, "High-power ultrafast fiber laser systems," IEEE J. of Sel. Top. in Quantum Electron. 12, 233-244 (2006). [CrossRef]
  2. F. Röser, J. Rothhard, B. Ortac¸, A. Liem, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, "131W 220 fs fiber laser system," Opt. Lett. 30, 2754-2756 (2005). [CrossRef] [PubMed]
  3. F. Röser, D. Schimpf, O. Schmidt, B. Ortac¸ K. Rademaker, J. Limpert, and A. Tünnermann, "90W average power 100 ?J energy femtosecond fiber chirped-pulse amplification system," Opt. Lett. 32, 2230-2232 (2007). [CrossRef] [PubMed]
  4. F. Röser, T. Eidam, J. Rothhard, O. Schmidt, D. N. Schimpf, J. Limpert, and A. Tönnermann, "Millijoule pulse energy high repetition rate femtosecond fiber chirped-pulse amplification system," Opt. Lett. 32, 3495-3497 (2007). [CrossRef] [PubMed]
  5. D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 219-221 (1985). [CrossRef]
  6. Y. H. Chuang, D. D. Meyerhofer, S. Augst, H. Chen, J. Peatross, and S. Uchida, "Supression of the pedestal in a chirped-pulse-amplification laser," J. Opt. Soc. Am. B 8, 1226-1235 (1991). [CrossRef]
  7. M. D. Perry, T. Ditmire, and B. C. Stuart, "Self-phase modulation in chirped-pulse amplification," Opt. Lett. 19, 2149-2151 (1994). [CrossRef] [PubMed]
  8. B. E. Lemoff and C. P. J. Barty, "Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses," Opt. Lett. 18, 1651-1653 (1993). [CrossRef] [PubMed]
  9. A. M. Weiner, "Femtosecond pulse shaping using spatial light modulator," Rev. Sci. Instrum. 71, 1929-1960 (2000). [CrossRef]
  10. D. Yelin, D. Meshulach, and Y. Silberberg, "Adaptive femtosecond pulse compression," Opt. Lett. 22, 1793-1795 (1997). [CrossRef]
  11. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997). [CrossRef]
  12. T. Brixner, M. Strehle, and G. Gerber, "Feedback-controlled optimization of amplified femtosecond laser pulses," Appl. Phys. B 68, 281-284 (1999). [CrossRef]
  13. K. H. Hong and C. H. Nam, "Adaptive pulse compression of femtosecond laser pulse using a low-loss pulse shaper," Jpn. J. Appl. Phys. 43, 5289-5293 (2004). [CrossRef]
  14. R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, "Thinning-out in optimized pulse shaping method using genetic algorithm," Rev. Sci. Instrum. 74, 2670-2674 (2003). [CrossRef]
  15. T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, "Feedback-controlled femtosecond pulse shaping," Appl. Phys. B 70, S119-S124 (2000). [CrossRef]
  16. K. Ohno, T. Tanabe, and F. Kannari, "Adaptive pulse shaping of phase and amplitude of an amplified femtosecond pulse laser by direct reference to frequency-resolved optical gating traces," J. Opt. Soc. Am. B 19, 2781-2790 (2002). [CrossRef]
  17. A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, "Adaptive control of pulse phase in a chirped-pulse amplifier," Opt. Lett. 23, 1915-1917 (1998). [CrossRef]
  18. A. Efimov and D. H. Reitze, "Programmable dispersion compensation and pulse shaping in a 26-fs chirped-pulse amplifier," Opt. Lett. 23, 1612-1614 (1998). [CrossRef]
  19. A. Efimov, M. D. Moores, B. Mei, J. L. Krause, C. W. Siders, and D. H. Reitze, "Minimization of dispersion in an ultrafast chirped pulse amplifier using adaptive learning," Appl. Phys. B 70, S133-S141 (2000). [CrossRef]
  20. G. Chériaux, O. Albert, V. Wänman, J. P. Chambaret, C. Félix, and G. Mourou, "Temporal control of amplified femtosecond pulses with a deformable mirror in a stretcher," Opt. Lett. 26, 169-171 (2001). [CrossRef]
  21. T. Tanabe, K. Ohno, T. Okamoto, M. Yamanaka, and F. Kannari, "Feedback control for accurate shaping of ultrashort optical pulses prior to chirped pulse amplification," Jpn. J. Appl. Phys. 43, 1366-1375 (2004). [CrossRef]
  22. S. Zhou, L. Kuznetsova, A. Chong, and F. W. Wise, "Compensation of nonlinear phase shifts with third-order dispersion in short-pulse fiber amplifiers," Opt. Express 13, 4869-4877 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-4869. [CrossRef] [PubMed]
  23. L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, "High energy femtosecond Yb cubicon fiber amplifier," Opt. Express 13, 4717-4722 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4717. [CrossRef] [PubMed]
  24. A. Chong, L. Kuznetsova, and F. W. Wise, "Theoretical optimization of nonlinear chirped-pulse fiber amplifiers," J. Opt. Soc. Am. B 24, 1815-1823 (2007). [CrossRef]
  25. L. Kuznetsova and F. W. Wise, "Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification," Opt. Lett. 32, 2671-2673 (2007). [CrossRef] [PubMed]
  26. L. Kuznetsova, A. Chong, and F. W. Wise, "Interplay of nonlinearity and gain shaping in femtosecond fiber amplifiers," Opt. Lett. 31, 2640-2642 (2006). [CrossRef] [PubMed]
  27. T. Schreiber, D. Schimpf, D. Muller, F. Röser, J. Limpert, and A. Tünnermann, "Influence of pulse shape in self-phase-modulation-limited chirped pulse fiber amplifier systems," J. Opt. Soc. Am. B 24, 1809-1814 (2007). [CrossRef]
  28. J. van Howe, G. Zhu, and C. Xu, "Compensation of self-phase modulation in fiber-based chirped-pulse amplifi-cation systems," Opt. Lett. 31, 1756 (2006). [CrossRef] [PubMed]
  29. G. Zhu, J. Edinberg, and C. Xu, "Nonlinear distortion free fiber-based chirped pulse amplification with self-phase modulation up to 2??," Opt. Express 15, 2530-2534 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2530. [CrossRef] [PubMed]
  30. D. N. Schimpf, J. Limpert, and A. Tünnermann, "Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum," Opt. Express 15, 16 945-16 953 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16945.
  31. F. He, H. S. S. Hung, J. H. V. Price, N. K. Daga, N. Naz, J. Prawiharjo, D. C. Hanna, D. P. Shepherd, D. J. Richardson, J. W. Dawson, C. W. Siders, and C. P. J. Barty, "High energy femtosecond fiber chirped pulse amplification system with adaptive phase control," Opt. Express (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5813. [CrossRef] [PubMed]
  32. 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, 1431-1433 (2008). [CrossRef] [PubMed]
  33. L. Lefort, J. H. V. Price, D. J. Richardson, G. J. Spühler, R. Paschotta, U. Keller, J. Fry, and A. R. Weston, "Practical low-noise stretched-pulse Yb3+-doped fiber laser," Opt. Lett. 27, 291-293 (2002). [CrossRef]
  34. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 3902-3905 (2004). [CrossRef]
  35. A. Präkelt, M. Wollenhaupt, A. Assion, C. Horn, C. Sarpe-Tudoran, M. Winter, and T. Baumert, "Compact, robust, and flexible setup for femtosecond pulse shaping," Rev. Sci. Instrum. 74, 4950-4953 (2003). [CrossRef]
  36. R. Storn and K. Price, "Differential evolution - A simple and efficient heuristic for global optimization over continuous space," J. Glob. Optim. 11, 341-359 (1997). [CrossRef]
  37. M. A. Ali, C. Khompatraporn, and Z. B. Zabinsky, "A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems," J. Glob. Optim. 31, 635-672 (2005). [CrossRef]
  38. H. Y. Fan and J. Lampinen, "A trigonometric mutation operation to differential evolution," J. Glob. Optim. 27, 105-129 (2003). [CrossRef]
  39. D. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, "Decrease of pulse-contrast in nonlinear chirpedpulse amplification systems due to high-frequency spectral phase ripple," Opt. Express 16, 8876-8886 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-8876. [CrossRef] [PubMed]
  40. B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, "Coherent control using adaptive learning algorithms," Phys. Rev. A 63, 063412/1-12 (2001). [CrossRef]

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