## Abstract

We report on the generation of linearly chirped parabolic pulses with 17-W average power at 75 MHz repetition rate and diffraction-limited beam quality in a large-mode-area ytterbium-doped fiber amplifier. Highly efficient transmission gratings in fused silica are applied to recompress these pulses down to 80-fs with an efficiency of 60%, resulting in a peak power of 1.7 MW. Power scaling limitations given by the amplifier bandwidth are discussed.

© 2002 Optical Society of America

## 1. Introduction

Today, the development of high repetition rate, high average power femtosecond lasers is pushed by real world applications. Micromachining of various solid targets makes special demands on the lasers systems regarding pulse duration, pulse energy and repetition rate [1,2].

Recently the direct generation of femtosecond laser pulses from passively mode-locked Yb-doped thin disk lasers with average output powers up to several 10 watts is reported [3,4].

Alternatively to the application of high power laser oscillators, amplifiers can be used to boost up the output of a low power ultrashort pulse oscillator to high average powers. An attractive approach is the application of fiber optical amplifiers.

The first fiber lasers were operated in the beginning of the sixties at wavelengths around one micron with output powers in the order of a few milliwatts. Owing to recent developments of reliable high brightness all solid state pump sources and the advances in fiber manufacturing technology these devices are no longer restricted to low-power operation. Their main performance advantages compared to conventional bulk solid-state lasers result from the combination of beam confinement and excellent heat dissipation. The generation of high power cw-radiation can be considered as a straightforward problem. Continuous-wave powers of well above 100 W [5,6] have been achieved with the cladding pump technique [7]. Along with the inherent compactness and stability fiber based systems offer in many fields a significant higher potential for applications than their bulk solid-state laser counterparts.

Ytterbium-doped fibers provide several key advantages regarding the amplification of short optical pulses. The gain bandwidth supports pulses as short as ~30 fs, the huge saturation fluence allows the generation of millijoule pulses [8] and the high optical pumping efficiencies (often greater than 80% [9]), make an ytterbium-doped fiber amplifier to more than an alternative to conventional solid-state laser systems.

However, power and energy scaling of ultrafast single-mode fiber amplifiers is restricted due to nonlinear pulse distortions, which are enforced by the large product of intensity and interaction length inside the fiber core.

This limitation can be overcome by sufficient pulse stretching in the time domain and the enlargement of the mode-field diameter of the fiber to reduce the nonlinear effects such as stimulated Raman scattering (SRS) and self-phase modulation (SPM). The application of this technique leads to a chirped-pulse amplification (CPA) system based on either a low-numerical aperture large-mode-area fiber [10] or a multimode large-core fiber [11], where power scaling is limited by the maximum acceptable phase distortion due to self-phase modulation. Using this technique pulse energies in the range of 100 μJ to 1 mJ at sub-picosecond pulse duration are demonstrated at repetition rates of typically less than 10 kHz [12,13]. Basically, average powers in the 100 W regime are possible, because there are no limitations in scaling the repetition rate to several 100 kHz.

However, the nonlinearity can be used to control the propagation of the pulses in a high power fiber amplifier. In combination with normal dispersion and gain linearly chirped parabolic pulses can be created. In this contribution, we report on the generation of 17-W parabolic pulses at a center wavelength of 1060 nm. A diffraction grating compressor based on novel type transmission gratings in fused silica is applied to remove the linear chirp, resulting in 80-fs pulses with 10.2-W average power. Complete spectral and temporal analysis is carried out using numerical simulations, where excellent agreement with the experimental results is achieved.

## 2. Parabolic pulse amplification in optical fibers

The numerical investigation of the nonlinear Schrödinger equation (NLSE) with gain, which is given by

reveals that the interplay of normal dispersion, nonlinearity (self-phase modulation) and gain produces a linearly chirped pulse with a parabolic shape, which resists optical wave breaking [14,15]. The linear chirp can be efficiently removed, allowing high-quality pulse compression. In equation (1) A(z,T) is the slowly varying pulse envelope in a retarded time frame, β_{2} is the group-velocity-dispersion parameter, γ is the nonlinearity parameter and g is the gain coefficient. For an amplifier with constant distributed gain, an exact asymptotic solution has been found that corresponds to a parabolic pulse that propagates self-similar [16]. The asymptotic pulse characteristics are not influenced by the shape or width of the input pulse. Only the initial pulse energy determines the final pulse amplitude and width.

We numerically analyze the NLSE using the standard split-step Fourier method [17]. Figure 1 shows the evolution of a Gaussian input pulse at 1060 nm center wavelength with realistic parameters of a 10-m long ytterbium-doped fiber amplifier. The initial pulses have a duration of ΔT_{0} = 500 fs and energy E_{i} = 400 pJ. The fiber amplifier provides a gain of 30 dB (g = 0.69 m^{-1}), β_{2} = 0.02 ps^{2} m^{-1} and γ = 5.0∙10^{-4} W^{-1}m^{-1}. The final parabolic pulses have a pulse duration (FWHM) of 7.1 ps and a spectral width of 50.9 nm, corresponding to a time-bandwidth product of 91.5. The evolution of the pulse envelope and the spectral distribution as a function of the propagation distance is illustrated in figure 2 in a 2D-color map.

Since the experimental demonstration of self-similar parabolic pulse propagation and amplification in optical fibers [18] the generation of 100-fs pulses with an average power of 5.0 W (13 W before compression) is reported [19]. Even pulses durations as shorts as 52-fs at low average output powers are achieved applying Yb-doped fiber based parabolic pulse amplification and additional third-order dispersion compensation [20].

## 3. Experimental setup

The experimental setup of our high-power femtosecond fiber amplifier system is shown in figure 3. The system consists of a passively mode-locked, diode-pumped solid-state laser system, a diode-pumped ytterbium-doped fiber amplifier and a diffraction-grating compressor based on transmission gratings.

As a femtosecond seed source a Nd:glass laser system is applied, which is based on a semiconductor saturable absorber mirror. The laser is running at 75 MHz repetition rate, producing pulses as short as 180 fs at ~1060 nm center wavelength and an average power of 100 mW. An optical isolator is used to avoid feedback from the high power fiber amplifier in the oscillator.

The high power amplifier is constructed using 9-m length of low-NA large-mode-area fiber with a 30-μm diameter, 0.06-NA step-index ytterbium-doped core, a 400-μm D-shaped inner cladding with NA = 0.38. The ytterbium doping concentration is 500 ppm (mol) Yb_{2}O_{3}. The calculated mode-field-diameter of the fundamental mode in this fiber is about 23 μm. This fiber has a V-parameter of about 5 and supports 4 transverse modes. However, coiling the fiber in a radius of less than 10 cm the bending losses discriminate the higher order transversal mode and only the fundamental mode is guided and amplified. We measured a M^{2}-value of 1.1 at highest output power. As pump source a pigtailed diode-laser emitting at 976 nm is employed.

In order to allow an optimized recompression of the parabolic pulses after the amplifier stage, in our group transmission gratings have been designed and manufactured.

Starting point for the design problem is a binary grating in fused silica illuminated by a plane monochromatic TE-polarized wave with a wavelength of λ = 1060 nm at the Littrow-angle, which is about 41.5° for a grating pitch of 800 nm. With regard to this setup the diffraction efficiencies were obtained by rigorous numerical calculation using the Fourier Modal Method [21]. Figure 4 shows the –1^{st} transmitted order efficiency subject to groove depth and duty cycle, which is defined as the linewidth-to-grating pitch-ratio in this case, in a 2D-color-map. For a groove depth of 1.54 μm and a duty cycle of 0.45 a diffraction efficiency of 97% is accomplished. The calculation of the diffraction efficiency as a function of wavelength and angle of incidence is shown in figure 5. A diffraction efficiency of more than 94% is obtained for a wavelength range from 1010 nm to 1110 nm and a deviation from the Littrow angle of ±4°. These results confirm that these gratings are highly suitable for broad-bandwidth ultrashort pulse compression. Besides their high efficiency these transmission gratings possess an order of magnitudes higher damage threshold compared to conventional gold-coated diffraction gratings. Therefore the compression of high power or high energy pulses is much more relaxed.

The fabrication of the large area gratings (10×60mm) is done by electron-beam-lithography. The following etching process (Reactive Ion Etching and Reactive Ion Beam Etching) results in a binary grating with the expected parameters. A scanning-electron microscope image is illustrated in Figure 6.

## 4. Experimental results

#### 4.1. Initial conditions

Design criteria for parabolic pulse amplifiers are published in Ref. 16 including the optimum initial pulse duration ΔT_{i,opt} and pulse energy E_{i,opt} to ensure the fastest convergence to the parabolic regime. At given parameters β_{2}, γ, and g the optimum initial conditions are given by

At the parameters β_{2} = 0.02 ps^{2} m^{-1} and γ = 5.0∙10^{-4} W^{-1}m^{-1} of the ytterbium-doped 30-μm LMA fiber amplifier with a gain of approximately 30 dB (g = 0.76 m^{-1}) and an input pulse energy of 133 pJ (10 mW, 75 MHz) the optimal initial pulse duration is 315 fs. At the given 180 fs of the Nd:glass oscillator the optimal initial pulse energy (Equation (3)) is 25 pJ. That reduction of the seed power would not saturate the high power fiber amplifier and increasing amplified spontaneous emission would be observed. Furthermore a characteristic propagation distance corresponding to the position where the pulse reaches the parabolic regime can be formulated [16]. Following this analysis, even with our non-optimized initial conditions the pulse enters the parabolic regime in the 9 m long fiber amplifier.

#### 4.2. Output characteristics of the Yb-doped amplifier

Seeding the power amplifier fiber with 10 mW of femtosecond pulses from the Nd:glass oscillator enables us to produce up to 20 W average output power. The slope efficiency is 63% with respect to the absorbed pump power (shown in figure 7).

The combined action of gain, nonlinearity and normal dispersion ensures the generation of linearly chirped parabolic pulses which experience large temporal and spectral broadening (see figure 1 and 2). Figure 8 shows the measured autocorrelation trace of the output pulses at 17 W average output power. The pulses possess an autocorrelation width of 6.4 ps corresponding to a pulse duration of 5.6 ps assuming a parabolic shape.

The spectral broadening in the parabolic amplifier is illustrated in figure 9 showing the emitted spectrum at different output powers in a logarithmic scale. The conversion of the spectrum of the initial hyberbolic secant pulse to the spectrum of a linearly chirped parabolic pulse can be identified. At an average output power of 17 W the spectral width is increased to 40.5 nm, corresponding to a time-bandwidth product of 58.2.

#### 4.3. Simulation of the experiment

Numerical investigations of the nonlinear Schrödinger equation (NLSE) using the standard split-step Fourier method are carried out to simulate the pulse evolution in the experiment.

Figure 10 shows the calculated output spectra at pulse energies corresponding to the given average powers in figure 9. The comparison with the experimentally measured spectra shown in figure 9 reveals a good agreement. The simulated spectral width at 17 W average output power (~ 230 nJ) is 40.6 nm. The oscillations in the spectrum are associated with a deviation from perfectly linear chirp across the parabolic pulse [16].

The simulated pulse shape and temporal phase as well as the corresponding autocorrelation trace are shown in figure 11 and indicate the parabolic shape of the pulse. The calculated pulse duration is 5.9 ps and the autocorrelation width is 6.8 ps at 17 W average output power of the LMA fiber amplifier. The uncertainty of the knowledge of the nonlinearity and the dispersion parameter of the real fiber amplifier causes the minor deviation from the measured values. Nevertheless, the simulations confirm the experimental results with a good agreement.

#### 4.4. Pulse compression

Transmission gratings in fused silica are employed to remove the chirped of the high power parabolic pulses. The gratings are used under Littrow angle. Best compression is achieved at a grating separation of 8.5 mm. Figure 12 shows the measured autocorrelation trace of the recompressed pulses at an output power of 17 W of the fiber amplifier. We determined the FWHM pulse duration, τ_{p} =80 fs, by assuming a sech^{2} pulse shape, corresponding to a time-bandwidth product of 0.86. The low-intensity wings in the autocorrelation trace have their origin in uncompensated higher-order phase contributions of the fiber amplifier grating compressor setup.

The compressor efficiency in a double pass configuration is as high as 80 %, if the gratings are illuminated with TE-polarized light. This corresponds to an experimental diffraction efficiency of a single grating of 95 %. The degree of polarization of the fiber amplifier output is decreased to 63% (at 17 W output power) due to the inherent variation of birefringence in the large-mode-area fiber. Nevertheless, a compressor efficiency of 60 % is reached, resulting in an average power of 10.2 W after compression, corresponding to a pulse peak power of 1.7 MW.

The 80-fs pulse duration is confirmed by a FROG-measurement [22] based on second harmonic generation. Figure 13 (a) shows the measured FROG trace of the recompressed pulses at the highest output power. The retrieved intensity of the electric field and phase in the time domain is illustrated in figure 13 (b) The FROG error was with 0.0053 acceptably low. The corresponding phase is nearly constant over the central part of the pulse, indicating efficient chirp removal.

#### 4.5. Power scaling limitations

The output power is limited to 17 W average power in this experimental configuration due to the amplifier bandwidth of the ytterbium-doped fiber. If the spectral width exceeds the gain bandwidth due to spectral broadening induced by self-phase modulation in the fiber amplifier, the propagation of the parabolic pulse is distorted and deformations of the pulse shape and spectrum can be observed. Apparently the linear chirp of the parabolic pulse isn’t sustained beyond this limitation leading to a poor quality of the recompressed pulses. Figure 14 shows the autocorrelation trace of the recompressed pulses at an average output power of 20 W.

Furthermore, following our simulations the threshold of stimulated Raman scattering is expected at an output power of approximately 25 W at the given experimental conditions.

Further power scaling can be achieved by increasing the core diameter of the fiber amplifier. The maximum extractable energy scales with the square of the fiber-core diameter and stable transverse single-mode operation of fibers with core diameters up to 50 μm is already reported [13]. Using a properly designed 50-μm core fiber sub-100 fs pulses with an average output power in excess of 50 W are possible in a reliable and less complex concept than a fiber based CPA system.

## 5. Conclusion

In conclusion, we have investigated a very simple and reliable approach of fiber based high power short pulse generation. The propagation of the ultrashort pulses during the amplification is controlled by nonlinear effects. Linearly chirped parabolic pulses with an average power of 17 W and diffraction-limited beam quality are generated using an efficient ytterbium-doped large-mode-area fiber amplifier. These pulses are recompressed to 80 fs pulses duration with an efficiency of 60 % applying transmission gratings in fused silica. The obtained average power of 10.2 W corresponds to a pulse peak power of 1.7 MW. Numerical simulations using split-step Fourier method have shown excellent agreement with the experiment. Based on this calculation, by optimization of the mode-field diameter of the fiber amplifier a power scaling to the 50 W level is possible.

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