## Abstract

We report on the experimental demonstration of the control of the influence of nonlinearity in fiber-based chirped-pulse amplification (CPA) using active spectral amplitude shaping. By applying a liquid crystal spatial light modulator, the influence of the spectral profile on the recompressed pulse quality is experimentally revealed. The parabolic spectrum is experimentally determined to be very suitable for CPA-systems in which nonlinearity is present. The corresponding nonlinear phase contribution can be efficiently compensated by a conventional grating compressor. In a proof-of-principle experiment using an Yb-doped fiber-CPA-system, control at a B-integral as high as 16 rad is demonstrated. The method allows significant performance improvement of fiber-based chirped-pulse amplification.

©2007 Optical Society of America

## 1. Introduction

Fiber-based ultrafast sources are particularly appropriate to the generation of pulse-trains of high average power at high repetition rates [1]. Fiber-lasers feature a high single pass gain, excellent beam quality, very good heat dissipation, efficient diode-pumped operation, and the potential of all-fiber integrity. The diffraction-less propagation of light in modes of small effective area over the entire (long) length of the active fiber mainly causes these advantageous attributes but also enforces the occurrence of nonlinear effects. The combined effect of nonlinearity and dispersion during pulse amplification can result in a phase-profile of complicated shape and high magnitude; therefore, it is difficult to compensate for it. As a consequence, the quality of the pulse is severely degraded after the amplification unit. Particularly, this makes peak-power and energy scaling with short pulse fiber-based amplifier systems challenging.

A lower peak-intensity during amplification decreases the impact of nonlinear effects. The amplification of stretched pulses (i.e., with reduced peak-power) in short-length large mode-area fibers solves this problem to a great extend [2, 3]. In state-of-the-art fiber CPA-systems [4], the temporal stretching of the sub-picosecond pulses with dielectric gratings is limited to a few nanoseconds, and stretching beyond this range becomes increasingly impractical. Furthermore, the large-mode area (LMA) fibers of short length (i.e., ~1 m), such as the recently developed microstructured fibers [3, 5], are limited in core-diameters to less than 100 µm [6]. If energy-levels in the mJ-range are envisaged with fiber technology, new experimental techniques are required that control the influence of nonlinearity during amplification, and thus, offer further energy and peak-power scaling.

Spectral gain shaping in a nonlinear fiber CPA-system without active pulse shaping has already been investigated [7]. In this contribution we show experimentally that a parabolic spectral shape permits increase in ultrafast pulse quality after amplification in highly nonlinear fiber-based CPA-systems. A few aspects of the experimental setup have been covered in previous conference contributions [8, 9], though, in this article, we give a complete description of the novel experimental method and its underlying physical principle. Furthermore, we confirm the correctness of the experimental results by numerical modelling of the fiber CPA-system.

In our approach, the influence of the dominant nonlinear effect of self-phase modulation (SPM) is controlled via the intensity profile. This is possible, since the SPM in a fiber-amplifier (without dispersion) is directly proportional to the temporal intensity distribution [10]:

${\phi}_{\mathrm{SPM}}(L,T)=\gamma {\mid {A}_{\mathrm{in}}\left(T\right)\mid}^{2}\frac{\left[\mathrm{exp}\left(\mathrm{gL}\right)-1\right]}{g}.$

Where the fiber-nonlinearity parameter is denoted *γ*, and |*A _{in}*(T)|

^{2}is the temporal power-profile at the input of the fiber-amplifier,

*g*is the gain coefficient for an exponential law amplification,

*L*is the length of the amplifier. It is important to stress, that in the equation above it has been assumed that the pulse width does not increase substantially during amplification. If nonlinearity is present during amplification, the acquired nonlinear phase, and thus, the corresponding intensity-profile, has a strong influence on the recompressed pulse quality at the output of the fiber-based CPA-system. For the example of an initial sech^2 pulse, this is shown in Fig. 1. It can be seen that with an increase of accumulated nonlinear peak-phase shift (i.e., the B-integral [11]), the pulse-quality at the output of the fiber-based CPA-system decreases.

In our approach we actively modify the spectral amplitudes of the pulse to be amplified. Analogous to the Fraunhofer-approximation in paraxial optics of diffraction, for a strongly chirped pulse the spectrum is mapped into time-domain [12, 13, 14]:

${A}_{\mathrm{in}}\left(T\right)\u2254A(z,T)=\frac{1}{2\pi}\int d\Omega \tilde{A}\left(\Omega \right)\mathrm{exp}\left(i\frac{{\beta}^{\left(2\right)}}{2}{\Omega}^{2}z\right)\mathrm{exp}(-i\Omega T)\approx \frac{\mathrm{exp}(-i\frac{1}{2{\beta}^{\left(2\right)}z}{T}^{2})}{\sqrt{2\pi \left(-i\right){\beta}^{\left(2\right)}z}}\tilde{A}\left(\frac{T}{{\beta}^{\left(2\right)}z}\right).$

Where the second derivative of the propagation constant of the stretcher is denoted β^{(2)}, and Ω is the difference of the angular frequency with the center angular frequency. From the relation above follows that a modulation of the spectrum directly affects the temporal intensity distribution of the stretched pulse, too. Here, we suppose that the intensity distribution in time-domain exhibits the same profile as the spectrum. For this, we neglect higher-order dispersion during the stretching. This assumption is reasonable as the ratio of third-order to second-order phase is on the order of magnitude of ~10^{-2}. We also neglect higher-order contributions which arise from the mapping between the domains [13]. A parabolic spectrum at the end of the fiber-amplifier is very suitable as the corresponding nonlinear phase contribution can be efficiently removed using the second-order dispersion of the grating compressor.

It is important to emphasize that the approach described here allows the presence of nonlinearity, though, its impact is controlled. Our method differs significantly from the concept of parabolic amplification, in which the self-similar evolution of the pulse during amplification is a key aspect [15]. Chirped-pulse amplification allows the generation of higher pulse energies compared to parabolic (or self-similar) amplification. In our parabolic-spectrum- technique applied to a fiber-based CPA system, apart from gain-shaping, the pulse shape is rather constant during pulse propagation and short-length LMA fiber-amplifiers can be used. The method uses a pulse shaper, such as a liquid crystal spatial light modulator (LC SLM) [16], before the amplifier in order to pre-shape the spectrum so that a parabolic spectrum is produced at the end of the fiber-amplifier, which is embedded in the CPA configuration. Furthermore, an active adaptive element allows overcoming spectral modulations imposed by the stretcher unit (such as spectral grating efficiencies), e.g., and spectral gain-shaping effects. So, it’s a more reliable as well as versatile technique than simply precede the CPA part with a self-similar amplification to shape the spectrum into a parabola.

The amplification of asymmetric spectral amplitude profiles (generated by static filters) in a fiber-stretcher-grating-compressor CPA configuration showing a third-order-dispersion mismatch has been used for the compensation of the phase due to SPM [17, 18]. However, the approach lacks the flexibility of the method presented in this contribution: the total phase due to both second-order dispersion and third-order dispersion of the nonlinear CPA system has to be compensated by careful choosing of the right length of stretcher fiber and corresponding separation in the grating compressor as well as the peak-power of the pulse during amplification. Therefore, there exists only one ideal point of output power operation (corresponding to the right B-integral in order to balance the dispersion) for a chosen stretcher-compressor configuration. In contrast, the method described here allows accommodating large magnitudes of parabolic phase due to nonlinearity (i.e., corresponding to high B-integrals) by simply changing the separation between the gratings in the compressor. In particular, this works at all power levels in which the system operates in the CPA regime. So, it is quite simple to remove the total second-order phase (including the nonlinear contribution) of the system without putting a high premium on higher-order phase compensation.

Phase-only LC SLMs have been implemented in CPA-systems [19, 20]. In other experiments electro-optic phase-modulators were used in fiber CPA-systems [21, 22]. Though, the maximum phase-shifts of the devices are limited. With the present state of technology, phase-shaping cannot compensate for the high magnitudes of SPM in a CPA-system.

## 2. Experiment

A schematic of the experimental set-up is shown in Fig. 2. The master oscillator is a passively mode-locked Nd:glass laser delivering sech^{2} pulses with durations of 180 fs (FWHM) at a center-wavelength of 1060 nm. The average power of the laser output is 100 mW, and the repetition rate is 75 MHz. The output is coupled into 100 m length of passive polarization maintaining fiber (PM 980). The fiber serves two purposes. Firstly, the pulses are spectrally broadened to about 40 nm (FWHM) in the initial section of the fiber, and secondly, the pulses experience temporal stretching to about 160 ps due to group-velocity dispersion. The coupling efficiency into the fiber is about 80%. Behind the fiber-stretcher the liquid crystal spatial light modulator (LC SLM) is placed. It is set-up in a reflective configuration: at the reflection grating (1200 lpmm) the beam is spectrally fanned out, then, a cylindrical focusing mirror (f=0.3 m) images the spectral components onto a liquid crystal mask which is positioned in the Fourier plane. The mask comprises 640 stripes (each 100 µm wide and 1cm high). The light is reflected directly behind the mask by a mirror. Despite the capability to modify both the amplitude and the phase, amplitude shaping is solely employed. After double passage through this configuration, the beam is steered into the fiber-amplifier. The amplifier is an Ytterbium-doped double cladding fiber with a core-diameter of 4 µm (NA=0.17) and a 400 µm D-shaped inner cladding (NA=0.38). The length of the fiber is 24 m and the doping concentration is 7000 ppm. The long length and the small core-size of the fiber-amplifier were intentionally chosen in order to obtain high B-integral values at lower peak-powers in a proof-of- principle experiment. The fiber-amplifier is pumped by a pig-tailed diode lasing at 976 nm.

The amplified pulses are recompressed by a pairs of transmission gratings with a groove frequency of 1250 lpmm. The transmission through the grating-compressor (set up in double pass configuration) is about 80 %. The spectra of the pulses are monitored by an optical spectrum analyzer (OSA), and temporal features are measured with an autocorrelator. A computer is connected to both the OSA and the LC SLM. In an iterative procedure, the acquired spectrum serves as a feedback signal for the calculation of a transmission function at the LC SLM. This results in a spectrum that is similar to the profile of the aim function (parabola). In particular, the difference between the measured spectrum and a least-squares fit are used for the calculation of a new transmission function at the LC SLM. The hands-off iterative routine is needed since the effective gain of the fiber amplifier exhibits a spectral dependency. Typically, about 20 iterations are sufficient to obtain the desired spectral profile. As the intensity of the pulse is the highest in the last section of the fiber-amplifier, the influence of SPM on the pulse propagation is the most dominant there. Therefore, the spectrum is measured behind the fiber-amplifier, and the pre-shaping precisely affects the nonlinear evolution in the last part of the fiber-amplifier.

By spectral amplitude modulation with the LC SLM, it is possible to generate almost any desired spectral profile. To experimentally reveal the dependency of the self-phase modulation on the spectral profile of a chirped pulse, and thus, the influence of the profile on the recompressed pulse quality, we generate a Gaussian and a parabolic spectrum.

Firstly, the spectrum at the output of the fiber-amplifier is shaped to a Gaussian shape with 10 nm bandwidth (FWHM). The center wavelength at 1075 nm is chosen since the spectrally dependent gain of the amplifier fiber favors this spectral region. The spectral broadening before the stretching in the fiber-stretcher enables modulation of this spectral region. The generated spectrum at the end of the fiber amplifier is shown in Fig. 3(a). The corresponding pulse exhibits a duration of 45 ps. This value has been obtained from the autocorrelation of the uncompressed pulse. After applying spectral amplitude modulation (20% of the total spectral power is transmitted) with the SLM unit (with a throughput efficiency of ~35%) there is an average power of 3 mW left for the (coupled) seed of the fiber-amplifier. Then, the amplifier is operated at low and higher output powers by changing the pump-power while keeping the seed-power constant. The average output power values are 250 mW and 1500 mW, respectively.

Assuming an exponential law for the power amplification along the fiber, the B-integrals can be determined from the input and output power values as well as the pulse duration during amplification and the calculated effective mode-field area. We neglect dispersion during amplification for the estimation. The B-integrals, corresponding to the low and higher output power case, are calculated to be 3.5 rad and 16 rad, respectively. The exponential law assumption for the amplifier (operated in a counter propagating configuration) is confirmed by a numerical simulation based on the laser rate-equations. Figure 3(c) shows the measured autocorrelation trace of the amplified and recompressed pulses for the Gaussian case. In the experiment, the autocorrelations are optimized for shortest FWHM duration of the traces. The autocorrelation trace at a low output power (i.e., corresponding to B=3.5 rad) shows a minor pedestal which has its origin in uncompensated higher-order dispersion due to the fiber-stretcher- grating-compressor setup. Partly, it is also due to nonlinearity which already shows minor influence at a B-integral of 3.5 rad. The initial spectral broadening in the stretcher fiber also introduces a small contribution to the phase-mismatch. The autocorrelation trace at higher output powers (i.e., corresponding to B=16 rad ~5*π* rad) shows a considerable wing structure. It is important to note that the separation between the gratings of the compressor is optimized for best compression, i.e. a part of the phase due to SPM could be removed by the grating compressor. The observed degraded autocorrelation trace is in good agreement with numerical simulation of this fiber CPA-system which is shown in Fig. 4(c). This result clearly demonstrates that commonly used profiles lead to nonlinear frequency modulations, which can not be generally compensated for by simple grating compressors. From the autocorrelation, it can be concluded that energy is spread out from the central peak of the pulse to a pedestal, thereby severely degrading pulse contrast and lowering the peak-power.

In contrast, if a parabolic spectrum at the end of the fiber-amplifier is generated as shown in Fig. 3(b), the temporal contrast in the autocorrelation trace between the low and higher power case is almost preserved (Fig. 3(d)). The used parabolic spectrum has an identical bandwidth (FWHM) and spectral center as the Gaussian spectrum, and the same output power levels from the fiber-amplifier as in the Gaussian case are achieved. Compared to the Gaussian profile, a significant improvement of pulse quality is revealed at a B-integral of 16 rad. Because of the linear chirp imposed by SPM, an appropriate change of the grating separation distance results in a good removal of the phase which is due to nonlinearity.

The numerical simulations of the autocorrelation traces at the output of the CPA-system for the parabolic case are shown in Fig. 4(d). The temporal pulses are not transform-limited because of the residual third-order dispersion arising from the fiber-stretcher-grating-compressor configuration. If a grating-stretcher is used instead of a fiber-stretcher, the autocorrelations shown in Figs. 4(e) and 4(f) are expected for the Gaussian and parabolic spectrum, respectively. In the simulation, the grating separation distance of the compressor is again adjusted for best compression.

For the grating-stretcher CPA-system, the spectral phase-profiles of the recompressed pulses at the output of the fiber-based CPA-system are shown in Fig. 5. For the case of a Gaussian spectrum, strong residual spectral phase-modulations are present, see Fig. 5(a). These modulations are responsible for pre- and post-pulse in time-domain, and thus, explain the degradation of the autocorrelation trace shown in Fig. 4(e). To obtain transform-limited pulses at the output of the CPA-system, the Gaussian profile would require phase-shaping. At present, standard phase-shaping devices cannot compensate for such high magnitudes of phase-shifts. In our experiment amplitude shaping was solely applied. For the parabolic case of the spectrum, such strong spectral phase-modulation are not given. Therefore, the autocorrelation trace shown in Fig. 4(f) exhibits quality-improvement compared to the case of a Gaussian spectrum.

Though, the phase of the pairs of gratings is not completely parabolic but also exhibits a small relative contribution (i.e., ~10^{-2}) due to third-order dispersion. The numerical results of Fig. 4(f) show that the influence on the recompressed pulse-quality is negligible. In addition, the third-order phase contribution, which is present because of the change in separation distance that is needed so as to compensate for a parabolic phase due to nonlinearity, is small compared to the magnitude of the overall phase. In Fig. 5(b) it can be observed, although its influence on the recompressed pulse quality is small. For the compensation of higher magnitudes of B-integrals, it may play a role. In principle, a shape that deviates from a pure parabola could be generated, and it could induce a corresponding nonlinear phase which accounts for third-order dispersion mismatch. However, as the magnitude of this phase-mismatch is small, it is probably more reasonable to directly compensate for it by additional active phase-shaping. Other small residual phases, e.g. because of stretcher-compressor-phase- mismatch, could be also tackled via this approach.

It is important to note, that according to basic equations describing stimulated Raman scattering in optical fibers [10], the threshold for this nonlinear effect is expected at a peak-power evolution in the active fiber corresponding to a B-integral of about 25 rad. Thus, in our experiment we could control the influence of SPM at peak-powers approaching the Raman-limit.

## 3. Conclusion

In conclusion, we have experimentally demonstrated a novel technique that allows accommodating high magnitudes of nonlinear phase in a fiber-based CPA-system while preserving temporal contrast at the output. The spectrum of the stretched pulse is shaped to a parabolic profile and the grating compressor of the fiber-based CPA-system is used for the compensation of the resulting, mainly parabolic phase due to self-phase modulation. With this technique, accumulated nonlinear phase shifts as high as 16 radians could be controlled. The influence of SPM could be controlled at peak-power evolutions inside the active fiber that approach the Raman-limit.

Potentially, the parabolic-spectrum-technique will be applied in fiber CPA-systems using much higher stretching ratios (~1000) and short-length LMA main fiber-amplifiers. Since the ratio of the B-integral and the magnitude of the phase produced by the stretcher is a key parameter, such systems will correspond to higher B-integrals that can be controlled as compared to this proof-of-principle set-up. While the presented parabolic-spectrum technique overcomes nonlinearity-limitations of conventional fiber-based CPA systems, peak-power scaling with such systems will still depend on the magnitude of second-order phase produced by the stretcher. By using the parabolic-spectrum-method, we anticipate the generation of high-quality ultrafast pulses at the energy level of mJ.

## Acknowledgment

This work has been supported by the onCOOPtics-project from the German Federal Ministry of Education and Research (BMBF). The authors also acknowledge support from the Gottfried Wilhelm Leibniz-Programm of the Deutsche Forschungsgemeinschaft.

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