1. Introduction

Femtosecond laser pulses are nowadays widely used for various applications, which are reviewed in detail in Ref. [1]. The therein presented fields of applications result in a strong interest especially in fiber based mode-locked lasers, since these laser sources show a compact set-up, an excellent beam quality and a high reliability. However, the output pulse energies of those lasers are typically limited to about 1 nJ, which are too low for many applications¹. In ² are used, where the pulses are dispersively stretched and chirped before amplification, reducing nonlinear effects in the amplifier fiber. A following compressor provides (nearly) bandwidth-limited output pulses, if its dispersion is well adapted for compensation of the formerly introduced and additionally accumulated chirp.

In addition to the dispersive stretcher, the dispersion of the amplifier influences the pulse chirp. Owing to the absorption and emission resonances of the Er³⁺-ions the inversion of the active ions influences the refractive index and the dispersion of these amplifier-fibers especially within the gain spectrum³. Furthermore, owing to the energy extraction during pulse amplification, the inversion in the active fiber decreases not only along the fiber but also along the pulse shape. In accordance, the refractive index of the fiber is changing along the pulse, modifying the initial pulse chirp. Thus, the chirp of the amplified pulse depends additionally on the energy extraction. Our experiments showed that in an Erbium-fiber-based CPA-system the influence of the inversion on the refractive index is about one order of magnitude larger than predicted by the analysis in the literature³.

2. Setup

The experimental set-up, shown in Fig. 1, consisted of three main parts: a mode-locked fs-pulse source, which included a dispersive fiber stretcher, an adjustable repetition rate pulse-picker and a preamplifier, a two stage Er³⁺-fiber amplifier and a grating compressor.

The pulse source was based on a home-made Er³⁺-fiber laser with a repetition rate of 60 MHz, which was mode-locked by nonlinear polarization rotation and operated in the soliton regime⁴. This laser source emitted pulses with a spectral bandwidth of 22 nm (FWHM), a center wavelength of 1559 nm and a pulse energy of 0.2 nJ. The emitted pulses were dispersively stretched with a telecom-grade fiber module in a double pass arrangement with an overall normal group velocity dispersion (GVD) of 89 ps² (D = -70 ps/nm) and a specified dispersion slope (β₃/β₂) of -27 fs. Thereafter, the pulse repetition rate was reduced by a telecom-grade fiber-coupled Mach-Zehnder modulator (> 20 dB extinction ratio). The signal was then amplified in an Er³⁺-fiber preamplifier, which mainly compensated the losses introduced by the fiber stretcher and the modulator. A following second synchronously driven modulator increased the extinction ratio and suppressed the amplified spontaneous emission out of the preamplifier efficiently. The resulting output signal spectrum was centered at 1557 nm and showed a bandwidth of about 10 nm (FWHM). The decrease in bandwidth was attributable to gain narrowing in the preamplifier. The pulse energy depended slightly on the repetition rate and varied between 1.4 nJ (at 607 kHz) and 1.8 nJ (at 200 kHz). This pulse energy variation was caused by gain saturation effects in the preamplifier, as the pump power remained constant. It should be noted that the previously described elements operated at the same settings during all measurements, except of a variation of the modulator pulse picking rate, which is mentioned later.

The two-stage Er³⁺-fiber amplifier, shown in Fig. 1, was pumped with a 1480 nm continuous-wave pump source with a maximum output power of 4.0 W. The pump light was distributed to the two amplifier stages by a fiber-coupler with a splitting ratio of 1:22. It was then launched contradirectionally into the amplifier stages by 1480/1550 nm WDM-couplers. The amplification of this two-stage amplifier was adjusted by the pump power.

The Er³⁺-fibers of the amplifier had a doping concentration of 2300 mol ppm Er₂O₃, a mode-field area of 16 μm² at 1550 nm wavelength and a dispersion D=-22 fs/(nm•m). Furthermore, numerical calculation of the signal intensity distribution from the refractive index structure of the fiber showed an area overlap of 53% between the signal mode and the Er³⁺-doped core⁵. In order to avoid modulation instabilities, only fiber elements with normal dispersion were used in this amplifier⁶. In order to avoid lasing and to reduce energy extraction by amplified spontaneous emission a two stage amplifier set-up was chosen. In addition, back reflections were suppressed at the amplifier input and between the two amplifier stages by optical isolators (> 40 dB).

 

Fig. 1. Schematic set-up of the two-stage Erbium-doped fiber amplifier and the grating compressor.

Download Full Size | PDF

The positively chirped output signal of the fiber amplifier was collimated and then temporally recompressed in a double pass grating compressor with a gold-coated holographic 1100 lines/mm grating pair. The group velocity dispersion of the compressor was adjusted by variation of the grating distance. Furthermore, the dispersion slope was varied between -31.1 fs and -24.2 fs at 1558 nm by changing the incident angle on the first grating between 53.8° and 51.7° ⁷. The “transmission window” of the compressor extended from 1553 nm to 1565 nm and was limited by the width of the second grating. Behind the compressor, the temporal shape of the signal was analyzed with an interferometric autocorrelator. Additionally, a small part of the signal was focused into a 6 mm long BBO crystal to investigate the influence of the grating separation on the second harmonic generation efficiency as a measure of pulse quality in addition to autocorrelation measurements.

3. Experimental results and analysis

By adjusting the compressor grating pair for minimum output pulse duration in respect of the interferometric autocorrelation function, it was observed that with increasing pulse energy the grating separation had to be reduced by more than 4%. The same relation was observed, if the grating distance was adjusted for maximum second harmonic generation in the BBO signal. However, the second harmonic signal was more sensitive to variations of the grating distance than the observed influence on the autocorrelation.

The influence of the output pulse energy and the repetition rate on the pulse chirp was then investigated in detail by adjusting the grating separation for each amplifier setting for maximum second harmonic signal, while simultaneously the autocorrelation function was measured. The resulting variation of the compressor GVD is plotted in the left diagram of Fig. 2 versus the input pulse energy for various repetition rates and a compressor dispersion slope (β₃/β₂) of -28.9 fs. It can be seen that the required (negative) GVD of the compressor for shortest pulse duration increased linearly with pulse energy and a slope of about 8•10⁵ fs²/μJ (D/μJ = -620 fs/(nm∙μJ)). This effect was independent of the pulse repetition rate, indicating that it was not attributable to thermal effects in the fiber or to the variation of the pump power. It should be noted that the compressor introduces an anomalous GVD (β₂ < 0)⁷. Thus, the positive slope in this figure corresponds to a reduction of the absolute value of the compressor dispersion with increasing pulse energy. Hence, the up-chirp of the amplified pulses before the compressor decreased with increasing pulse energy.

 

Fig. 2. Left diagram: Compressor dispersion for pulse compression versus amplifier output pulse energy for three different pulse repetition rates. The compressor dispersion was measured relative to the setting for the asymptotic limit for 0 μJ pulse energy. Right diagram: Amplifier output spectrum for output pulse energies of 0.9 μJ, 3.7 μJ and 6.3 μJ (200 kHz repetition rate).

Download Full Size | PDF

Variations of the dispersion slope between the above mentioned limits, in addition to the grating separation adjustment, showed the lowest pulse distortions in the autocorrelation in combination with a compressor dispersion slope of -28.9 fs. This indicated that the best compensation of the quadratic pulse chirp (corresponding to third order dispersion compensation) was achieved for this compressor setting. During these experiments it was also confirmed, that the compressor adjustment for maximum second harmonic generation efficiency corresponded to the shortest output pulse durations. It should be noted that for pulse energies above about 5 μJ the second harmonic maximum and the pulse width minimum for the autocorrelation function became less definite.

The right diagram in Fig. 2 shows the amplifier output spectra for output pulse energies between 0.9 μJ and 6.3 μJ. It can be seen from these spectra that with increasing pulse energy the mean wavelength increased slightly from 1558.7 nm (at 0.9 μJ) to 1560.1 nm (at 6.3 μJ). Variation of the repetition rate showed that the output spectrum depended only on the output pulse energy and not on the repetition rate. Furthermore, it can be seen in Fig. 3 that the output spectrum showed spectral modulations, whose amplitude increased with pulse energy. These spectral modulations are attributable to nonlinear effects in the amplifier fiber and similar modulations were also observed in Yb³⁺ based chirped pulse fiber amplifiers⁸. The origin of these spectral modulations was discussed in more detail in reference [5] and is out of the scope of this publication.

 

Fig. 3. Left diagram: Interferometric autocorrelation for an output pulse energy of 0.9 μJ. The compressor was adjusted for a dispersion slope of -28.9 fs and minimum pulse duration. Right diagram: Pulse duration versus pulse energy.

Download Full Size | PDF

The left right diagram in Fig. 3 shows the resulting autocorrelation at a pulse energy of 0.9 μJ and a dispersion slope of -28.9 fs. The measured autocorrelation width of 1350 fs corresponds to a pulse width of 810 fs, assuming a Gaussian pulse shape⁹. For the pulse width, a bandwidth limit of 670 fs was calculated by a Fourier-transform of the spectrum (at 0.9 μJ) within the transmission bandwidth of the compressor. Hence, the compressed pulses were near to their bandwidth limit. With increasing pulse energy, the minimal pulse duration increased up to 960 fs, despite of the compressor readjustment (right diagram in Fig. 3). Furthermore, the autocorrelation showed a slight pedestal outside the interference region. This indicated additional higher order phase distortions introduced by nonlinear effects, which also caused the spectral modulations mentioned above.

The observed decrease of the pulse up-chirp with increasing pulse energy cannot be attributed to self-phase modulation in the amplifier fiber, as self-phase modulation increases the up-chirp over a large central region of the pulse. Furthermore, the intensity varies “slowly” due to the highly dispersive pulse stretching which strongly reduces the self phase modulation induced chirp, which is proportional to the intensity variations. For that reason, the calculated self-phase modulation induced chirp was below 1000 fs², despite of pulse peak powers of up to 10 kW. Furthermore, it is worth to mention that self-phase modulation has only a small influence on the output spectrum, apart from the spectral modulations, which is also a result of the “slow” intensity variation of the highly stretched pulse. Moreover, the observed variations of the signal chirp with pulse energy could not be attributed to the variation of the center wavelength, as this would require an uncompensated third order dispersion (TOD) of about 3•10⁹ fs³. However, such a highly uncompensated TOD can be excluded for the system set-up, as the dispersion slope of the compressor was adapted to the dispersive fiber stretcher (accuracy: ~2•10⁸ fs³). In addition, comparisons between the measured autocorrelation and the autocorrelation computed from the measured spectrum for such quadric chirps verified that such a high TOD would be in contradiction to the measured autocorrelation.

The decrease of the Er³⁺-inversion during pulse amplification results in a change of the refractive index (even for a constant wavelength) as well as of the GVD of the gain fiber, as previously mentioned. Both influence the pulse chirp as they affect the propagation velocity along the chirped pulse. The propagation velocity of the short wavelength pulse tail of a positively chirped pulse is increased relative to the long wavelength pulse front if the refractive index decreases during pulse amplification. Thus, a decrease of the refractive index with pulse energy extraction has the same effect on a positively chirped pulse as anomalous dispersion. The measured dispersion effect of 620 fs/(nm•μJ) requires therefore a refractive index decrease of 3.5•10^-4 per extracted μJ in the 4.2 m long Er³⁺-fibers. For this calculation, the pulse shape was approximated by a linearly chirped square pulse with 8 nm spectral bandwidth (calculation: 620 fs/(nm•μJ) • 8 nm = 3.5•10^-4/μJ • 4.2 m / c, c as the speed of light). A linear variation of the GVD with inversion, and therefore with extracted pulse energy, would mainly result in a variation of the quadratic chirp, which corresponds to the TOD. Nevertheless, a Taylor-expansion at the mean wavelength (= pulse center) would show a variation of the fiber GVD for such a quadratic chirp. Thus, the observed GVD variation would require an increase of the fiber GVD in the 4.2 m long fiber by more than 150 fs/(nm•m•μJ).

Based on the fiber data it was calculated, that an output pulse energy of 1 μJ corresponds to an energy extraction of about 0.5% of the Er³⁺-ions in the final (two-stage) amplifier. The corresponding resonance-based effects were calculated according to the equations given in reference [3] for quasi-static Er³⁺-population distributions by taking into account the doping concentration as well as the doping and the mode-field distribution of the used fiber. These computations predicted a refractive index change of about -7•10^-6/μJ and a GVD variation of -0.1 fs/(nm•m•μJ). Thus, the calculated refractive index change is more than one order of magnitude below the experimentally observed results. The calculated change of the GVD is ⁶, which were not seen in the output spectrum. This leads to the conclusion that the measured influence of the pulse energy on the pulse chirp was (mainly) caused by the nonlinear refractive index change in the Er³⁺-fiber and not by the change of the fiber GVD.

The above used equations from reference [3] include the assumption that the population distribution within each laser level is in thermal equilibrium. However, the large energy extraction of these pulses may cause (relatively) large deviations from the thermal equilibrium within the energy levels. The duration of these chirped pulses was about 0.5 ns, as they were stretched by 70 ps/nm and had a bandwidth of about 8 nm. Furthermore, at the signal wavelength of about 1.56 μm laser emission terminates into a sublevel above the lowest energy level of the ground state. Hence, the energy extraction may result in a large overpopulation of this sublevel during pulse amplification, relative to thermal equilibrium. Moreover, the previously mentioned equations of reference [3] showed a more than one order of magnitude larger influence of the Er³⁺-inversion on the refractive index at about 1.53 μm, which corresponds to absorption or emission from the lowest ground state sublevel, which is mainly populated at inner band thermal equilibrium. Hence, such a strong resonance can also be expected at longer wavelengths if sublevels above the lowest ground state become more populated. Therefore, the deviation of the population distribution within each laser level from thermal equilibrium probably causes a much larger refractive index change during pulse amplification than previously calculated. However, for a detailed quantitative description the time constants for the transitions between the various sublevels of the ground state must be determined with fs accuracy by additional experiments. Based on these data and the transition probabilities between the various sublevels the time dynamic of the refractive index and the material dispersion during pulse amplification could then be calculated based on the Cramers-Kronig relation.

4. Conclusion

In conclusion, we have shown that the nonlinear dispersion effects on chirped pulses at 1.56 μm in an Er³⁺-fiber amplifier are about one order of magnitude larger than theoretically predicted. Thus, for optimum pulse compression high pulse energy CPA-systems based on Er³⁺-doped fibers must be either operated at constant pulse energy or require an adaptation of the compressor or the stretcher dispersion for different output pulse energies. Further experiments are necessary in order to determine the dependence of the nonlinear dispersion on the pulse duration, the signal wavelength and the gain fiber parameters. In addition, it should be analyzed, if similar effects are relevant for fs-CPA systems based on fibers with other active dopants (e.g. Nd³⁺, Yb^{3+ 10, 11}).