1 Introduction

Femtosecond lasers with sub-100 fs short pulse duration, which can provide high time resolution, are ideal tools for many scientific applications such as ultrafast spectroscopy [1], nonlinear imaging [2], and micromachining [3]. Yb fiber lasers have been employed to obtain such short-duration pulses in the past several years. Among various types of femtosecond lasers, Yb fiber lasers are particularly attractive due to their unique advantages such as high gain, broad gain bandwidth, excellent beam quality, compactness and robustness.

To date, femtosecond Yb fiber lasers are mostly configured in the architecture of master-oscillator-power-amplifier. To scale up average power and pulse energy of femtosecond Yb fiber lasers, there are two main amplification techniques: chirped pulse amplification (CPA) and nonlinear amplification. Fiber CPA is a mature linear amplification technique [4]. Yb fiber lasers based on CPA are the most widely used industrial femtosecond lasers. With large-mode-area (LMA) Yb-doped rod fiber, a single-channel fiber CPA system allows the generation of pulses with average power above 100 W and pulse energy at several-hundred-microjoules level [5,6,7,8]. However, compressed pulses of Yb fiber CPA systems are hardly shorter than 100 fs due to gain narrowing and residual dispersion mismatch between the stretcher and the compressor. An alternative approach is to utilize the nonlinear amplification methods. Three main nonlinear amplification methods for fiber amplifiers have been developed: self-similar amplification [9,10,11,12,13,14,15,16,17,18], pre-chirp managed amplification [19,20,21] and gain-managed amplification [22, 23]. With nonlinear amplification methods, amplified pulses can accumulate enough phase shift and spectral bandwidth can be significantly broadened beyond the gain bandwidth of Yb fiber. As a result, amplified pulses can be compressed to well below 100 fs.

Among these nonlinear amplification methods, self-similar amplification is demonstrated earliest and possesses some unique characteristics. In a self-similar amplifier, amplified pulses can evolve into linearly chirped pulses with a parabolic temporal profile due to the cumulative action of normal dispersion, self-phase modulation (SPM) and gain. The asymptotic parabolic pulses propagate self-similarly in the fiber amplifier, experiencing temporal and spectral broadening. The spectral broadening can ensure a compressed pulse duration much shorter than the original seed. Up to now, using a self-similar amplifier as the main amplification stage allows generation of several-hundred-nanojoules-level pulses with a compressed duration of shorter than 100 fs [12,13,14,15,16].

Besides acting as a main amplifier, a self-similar amplifier can also be used in an earlier amplification stage to generate parabolic seed pulses for fiber CPA system [17, 18]. In a high-energy fiber CPA system, the temporal shape of seed pulses plays an important role in improving the recompressed pulse quality. As amplified pulse energy increases, accumulated nonlinear phase, which is defined as B integral, leads to side-pulse and pedestal structure of recompressed pulses, thereby reducing the pulse peak power and degrading the pulse contrast [24]. This effect can be significantly mitigated by employing parabolically shaped pulses as initial seeder. D. N. Schimpf et al. have demonstrated that parabolic pulses propagating in a fiber amplifier can tolerate a B-integral up to 16 rad and still be recompressed to high-quality femtosecond pulses, which cannot be achieved if the initial seed pulses have other temporal shapes such as Gaussian [25]. In order to generate parabolic pulses, a spectral domain modulation approach based on a liquid crystal spatial light modulator has been employed [25, 26]. Compared with that approach, using self-similar amplification is natural and simple. Parabolic intensity profile, linear chirp and broad spectral bandwidth are intrinsic merits of a self-similar amplifier, and these characteristics are exactly what an optimal CPA seed laser requires. Moreover, the self-similar amplifier and the fiber CPA can be constructed in an all-in-fiber format, which makes the system robust and alignment-free.

A self-similar amplifier can act as a parabolic pulse shaper or a main amplifier. In both cases, to achieve self-similar amplification in gain fiber, input pulse energy and pulse duration should be controlled carefully, and the parameters of the amplifier should be chosen properly. According to the self-similar amplification theory [9,10,11], only the input pulse energy determines the amplitude and width of the asymptotic parabolic pulse solution, and for a fixed pulse energy, the rate at which the pulse evolves to the parabolic regime depends strongly on the input pulse duration. To ensure the fastest convergence to the parabolic regime, the optimal input pulse duration ∆Tin,opt is given by.

$$\Delta T_{{\text{in,opt}}} = 3 \cdot \frac{{\left( {\frac{{\gamma \beta_{2} }}{2}} \right)^{1/3} \cdot U_{{{\text{in}}}}^{1/3} }}{g^{2/3}}.$$
(1)

Here, β2 is the group velocity dispersion parameter, γ is the nonlinearity parameter, g is the gain coefficient, and Uin is the input pulse energy [10].

We use Eq. (1) to determine the optimal input pulse duration for the two categories of self-similar amplifiers: (a) a self-similar amplifier acting as a parabolic pulse shaper. In this case, the self-similar amplifier locates after the fiber oscillator, and before the stretcher of the following CPA system. Uin is relatively low (0.1–1 nJ), and the self-similar amplifier usually employs standard single-mode (SM) Yb fiber [17, 18]. And, (b) a self-similar amplifier acting as a main amplifier. In this case, to scale up the pulse energy, Uin is relatively high (0.1–10 nJ, or even higher), and the self-similar amplifier usually employs LMA Yb fiber [12,13,14,15,16]. In Fig. 1, by using Eq. (1), we plot ∆Tin,opt as a function of gain g for the two categories of self-similar amplifiers. We consider a typical 6-m self-similar amplifier with a practical gain ranging from 10 to 30 dB (g = 0.38–1.15 m−1). For standard SM Yb fiber, β2 = 0.023 ps2/m, and γ = 4.2 × 10–3/W/m. For typical LMA Yb fiber with core/clad diameters of 30/400 μm, β2 = 0.023 ps2/m, and γ = 0.5 × 10–3/W/m. In Fig. 1, calculated ∆Tin,opt for category (a) is represented by red area, and ∆Tin,opt for category (b) is represented by blue area. For reported works in which input and amplifier parameters are clearly presented [12,13,14,15], we have calculated the ∆Tin,opt and all of them fall into the corresponding area (orange points in Fig. 1).

Fig. 1
figure 1

Calculated optimal input pulse duration ∆Tin,opt as a function of amplifier gain g

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Figure 1 shows that the optimal input pulse duration for self-similar amplifiers with realistic parameters is several hundred femtoseconds, or even longer than 1 ps. However, nowadays, most of these input lasers come from commercial or homemade femtosecond Yb fiber oscillators or Yb fiber amplifiers, whose compressed pulse durations are usually in the sub-200 fs region or even shorter, owing to the broad gain bandwidth of Yb fiber. Therefore, the optimal input pulse duration for a self-similar amplifier is usually several times longer than the compressed pulse duration of commonly used femtosecond Yb fiber lasers.

In this work, we propose a simple and practical preshaping method for optimizing input pulses of a self-similar amplifier. We use a narrow bandpass filter (NBF) to clip the overbroad spectrum of input pulses, hence the pulse duration can be lengthened several times simultaneously. With the input pulse duration approaching the optimal value, the pulses can evolve to the parabolic regime more efficiently. We have experimentally demonstrated the preshaping method by achieving self-similar amplification in standard single-mode Yb fiber. The spectral bandwidth of amplified pulses is broadened to 42.6 nm, and the compressed pulse duration of 72.7 fs is much shorter than that of the oscillator pulses (118.2 fs). Previous input optimization methods for a self-similar amplifier are mainly focused on delicate pre-chirp control of input pulses [16, 18]. Compared with that, the spectrum-filtering method here, which is only based on the design criteria for self-similar amplifiers published in Ref. [10], is more straightforward. The laser source presented here can be used in applications that demand sub-100 fs short pulses, or it may be used as a parabolic pulse shaper to seed high-power CPA systems.

2 Experimental setup

The experimental setup of the fiber laser is illustrated in Fig. 2. A typical passively mode-locked Yb fiber oscillator is used to generate input pulses for the amplifier. All the fibers employed in the whole laser system are single-mode, single-clad, polarization-maintaining (PM) fibers. The homemade oscillator is constructed in a linear cavity configuration. A commercial SESAM (Batop GmbH) acts as one cavity end mirror and a CFBG (Teraxion) acts as the other. The SESAM ensures the self-starting of the mode-locking process and maintains the stability of the laser pulses against perturbations. Here, a SESAM chirp, an aspheric lens and a fiber collimator are integrated into one fiber component, which is compact and robust. The CFBG provides a negative dispersion of – 0.14 ps2 which compensates for the normal dispersion of the silica fiber of one cavity round trip. Meanwhile, the CFBG with a reflectivity of 22% serves as an output coupler for the oscillator. Besides that, the CFBG has a full width at half maximum (FWHM) reflection bandwidth of 25.8 nm centered at 1029 nm and acts as a spectral filter in the pulse shaping process. The Yb-doped fiber (PM-YSF-HI-HP, Nufern) is core-pumped by a 976-nm single-mode laser diode. Single-polarization operation is ensured by using a micro-optic polarizer inside the cavity. To prevent any disturbance due to back reflections, an isolator with 45 dB isolation is added to the CFBG output port.

Fig. 2
figure 2

Experimental Setup. SESAM semiconductor saturable absorber mirror, Yb-HI Yb-doped fiber with high Yb concentration, Yb-LO Yb-doped fiber with low Yb concentration, WDM-1, WDM-2 wavelength division multiplexers, CFBG chirped fiber Bragg grating, LD-1, LD-2 laser diode, ISO isolators, Col-1, Col-2, Col-3 collimators, TG-1, TG-2, TG-3, TG-4 transmission gratings, HR1, HR2, HR3, HR4 high reflective mirrors, NBF narrow bandpass filter, HWP half-wave plate

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To verify the pulse preshaping method, we only use two commonly used free-space optical components: a grating pair and a NBF. They are inserted between PM pigtailed collimator pair Col-1 and Col-2. Most existing self-similar amplifiers employ transform-limited (TL) pulses or nearly TL pulses as the input [12,13,14,15,16,17,18]. It is difficult to achieve self-similar amplification when highly chirped pulses are injected into the fiber amplifier. The oscillator emits chirped pulses due to its all-fiber configuration, therefore, we first compress the pulses to TL duration with a pair of transmission gratings TG-1 and TG-2 (1600 line/mm, LightSmyth). After the grating compressor, we use a NBF with a FWHM bandwidth of 3.9 nm and a center wavelength of 1030 nm (LL01-1030, Semrock) to clip the overbroad spectrum of the compressed pulses.

After the pulse preshaping components, the laser pulses are received by the collimator Col-2. The Col-2 is mounted on a 3-axis rotation mount paired with XYZ translation stages for coupling adjustment. A half-wave plate is placed before the Col-2 to tune the polarization direction of the laser coupling into the PM fiber. A 90:10 coupler (10% out) is placed after the Col-2 to monitor the received power and spectrum. Then, the pulses are injected into the fiber amplifier. The fiber amplifier consists of two gain fiber sections with different Yb concentration which are pumped by a same 976-nm single-mode laser diode. The first gain fiber is 40 cm of highly Yb-doped fiber with a core diameter of 6 μm and core pump absorption of 250 dB/m at 976 nm (PM-YSF-HI-HP, Nufern). It is used to pre-amplify the pulse energy without changing spectral and temporal characteristics of the pulses, compensating for the energy attenuation introduced by the pulse preshaping components. The second gain fiber, which has a relatively low Yb concentration and pump absorption of 80 dB/m at 976 nm (PM-YSF-LO-HP, Nufern), is used to realize self-similar amplification. The length of the second Yb fiber is fixed to be 8 m in the experiment. After the fiber amplifier, transmission gratings TG-3 and TG-4 (1000 line/mm, LightSmyth) are employed to remove the positive linear chirp of output parabolic pulses.

3 Experimental results

The mode locking of the fiber oscillator self-starts when we gradually increase pump power to 48.6 mW. As the pump power is increased to double pulsing threshold of 72.4 mW, we can observe multi-pulse unstable behavior. We fix the pump power at 53.6 mW and stable single pulse operation is obtained at 37 MHz. The average power of laser pulses emitted from Col-1 is 7.2 mW, corresponding to a pulse energy of 0.19 nJ. The 2.5-ps chirped pulses are firstly compressed by the grating pair TG-1 and TG-2. The incidence angle is 55.5° and the transmission gratings operate in Littrow configuration. Owing to the diffraction efficiency of a single grating of 94% for s-polarized laser, the efficiency of the double-pass grating compressor is as high as 76%. The laser power after the grating pair is 5.5 mW. The spectral and temporal characteristics of the compressed pulses are shown in Fig. 3. The spectrum is measured with an optical spectrum analyzer (Yokogawa, AQ6373B). As shown in Fig. 3a, it is centered at 1032 nm, with a FWHM bandwidth of 15.5 nm. We optimize the grating separation to minimize the dechirped pulse duration. The autocorrelation curve of the shortest pulses, shown in Fig. 3b, is measured by intensity autocorrelation (FR-103XL, Femtochrome). By assuming a sech2 pulse envelope, the FWHM pulse duration is 118.2 fs, corresponding to a time-bandwidth product of 0.52. After the nearly TL pulses go through the 3.9-nm NBF, the spectral bandwidth of the pulses decreases to 4.0 nm, as shown in Fig. 3a. Meanwhile the pulse duration increases to 480.9 fs, as shown in Fig. 3b. After the NBF, the laser power is 1.3 mW.

Fig. 3
figure 3

Pulse spectral and temporal characteristics before and after the NBF, a spectrum measured with spectrum resolution of 0.02nm, b pulse autocorrelation trace and fitting

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Here, we choose to use a 3.9-nm bandpass filter for two reasons. First, with the parameters of the second gain fiber Yb-LO and its achievable input pulse energy, from Eq. (1) we can estimate that the optimal input pulse duration ∆Tin, opt is longer than 1 ps. However, cutting off too much of the spectrum leads to severe pulse energy attenuation. Therefore, we do not choose to use a narrower one to lengthen the pulse duration to ∆Tin, opt. Second, there are fibers before the Yb-LO section. After filtering, the 480.9-fs TL pulses will not be severely stretched and highly chirped before entering the Yb-LO section. We can calculate that the 480.9-fs TL pulses will be slightly stretched to 512 fs due to material dispersion. Considering the material dispersion of fibers before the Yb-LO section, we do not choose to use a filter with a broader bandwidth.

To experimentally verify that pulse shaping is critical for realizing self-similar amplification, we have injected the 2.5-ps oscillator output chirped pulses into the Yb-LO fiber section directly and observed the experimental results without the pulse shaper. There are no free-space components, and we can splice the ISO and the WDM-2 together. The Yb-LO fiber is spliced with the WDM-2 and forward pumped by the LD-2. Figure 4 shows the spectrum of the amplified pulses at different pump power. As expected, the spectrum is only broadened due to SPM, but not parabolically shaped in the Yb-LO fiber. We have compressed the amplified pulses at 300-mW pump power by use of a grating pair. Due to the large SPM-induced nonlinear chirp, the pulses cannot be compressed to below 200 fs and there are obvious pedestals in the autocorrelation traces of the compressed pulses. Therefore, we can conclude that the amplification cannot enter the parabolic regime without pulse shaping.

Fig. 4
figure 4

Spectrum of the amplified pulses in the experiment without pulse shaper

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In the experiment with the pulse shaper, the power of the laser coupling into the Col-2 is 0.81 mW. After the coupler, 90% of laser enters the two-gain-section fiber amplifier. The pump laser is firstly absorbed by the first highly Yb-doped fiber, and the residual pump laser is used to pump the second Yb fiber. The first Yb fiber is short and the amplified pulse energy mainly depends on the length of the Yb fiber. Figure 5 plots the output power of signal laser and residual pump laser in the first Yb fiber section. With a pump power of less than 150 mW, the signal power increases significantly. With higher pump power, the signal power does not change much due to the limit of the Yb fiber length. At the maximum available pump power of 500 mW, the signal power reaches 17.9 mW, corresponding to a pulse energy of 0.48 nJ. The maximum residual pump power is 312 mW at the pump power of 500 mW. The pulse amplification in the first Yb fiber section operates in a low-nonlinearity regime because of the low pulse energy and short Yb fiber length. Notably, pulses after spectrum filtering have narrower spectral bandwidth and are barely affected by material dispersion when propagating in the fibers before the second Yb fiber section.

Fig. 5
figure 5

Output power of the amplified signal laser and the residual pump laser versus pump power in the first Yb fiber section

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Owing to the pulse preshaping and preamplification, the amplification of pulses quickly evolves into the parabolic regime in the second Yb fiber section. The pulses are parabolic shaped and experience temporal and spectral broadening during the self-similar amplification. After amplification, the picosecond parabolic pulses can be compressed to short pulses with sub-100 fs duration. Figure 6 shows the average power, spectral bandwidth and pulse duration of the compressed pulses versus pump power. Here, the pump power refers to the power of pump laser entering the second Yb fiber section. In this section, the pump laser with a maximum power of 312 mW can be almost completely absorbed, and the signal laser power increases monotonically with increasing pump power (red-triangle curve in Fig. 6). At the pump power of 312 mW, input signal laser power of 17.9 mW is amplified to 214 mW. After the grating pair, the power of compressed pulses is measured to be 182 mW, corresponding to a pulse energy of 4.9 nJ. The orange-square curve in Fig. 6 shows that the pulse spectrum broadens as the pump power grows. Broader spectral bandwidth can support shorter TL pulse duration, therefore the compressed pulse duration decreases with increasing pump power (green-circle curve in Fig. 6). The spectrum and autocorrelation trace of compressed pulses measured at different pump powers are shown in Fig. 7. The evolution of the spectrum towards an ideal parabolic shape is affected by the gain profile of the Yb-LO fiber. Input pulses have a narrowband spectrum centered at 1030 nm. Starting from 1030 nm, the spectrum broadens towards both longer and shorter wavelength sides because of the SPM effect. As shown in Fig. 7a–c, the spectrum extends towards the longer wavelength side further than the other side. The spectrum extension towards the shorter wavelength side is limited by the finite gain bandwidth of the Yb-LO fiber, therefore, as shown in Fig. 7c, the spectrum of the amplified pulses experimentally has spikes in the shorter wavelength side, which deviates from an ideal parabolic curve. The spectral shape on the longer wavelength side has a smooth edge which fits well with a parabolic curve, showing a typical feature of self-similar amplification. With maximum pump power of 312 mW, the broadest spectrum with a FWHM bandwidth of 42.6 nm is achieved, as shown in Fig. 7c. With a grating pair, linear chirp can be removed and compressed pulses of 72.7 fs are obtained (Fig. 7f), which is another typical feature of self-similar amplification. The slight pedestals in Fig. 7d–f are probably induced by the gain shaping effect or a small nonlinear chirp accumulated in the fiber amplifier. Compared with the 118.2-fs compressed pulses from the oscillator, amplified pulses have a much wider spectrum and are shortened to the sub-100 fs region.

Fig. 6
figure 6

Average power, spectral bandwidth and pulse duration of the compressed pulses versus pump power in the Yb-LO fiber section

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Fig. 7
figure 7

Spectrum and autocorrelation trace of compressed pulses at the Yb-LO pump power of 50 mW, 160 mW and 312 mW, a–c spectrum measured with spectral resolution of 0.02nm, d–f pulse autocorrelation trace and fitting

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To verify the excellent output laser quality of the fiber amplifier, we measure the radio-frequency (RF) spectrum and power stability of the 72.7-fs compressed pulses. Figure 8a shows the RF spectrum of the pulses measured with a photodetector (PDA10A2, Thorlabs) connected to a RF spectrum analyzer (N9320B, Keysight). The RF spectrum has a signal-to-noise ratio (SNR) higher than 70 dB, indicating that the pulses are stable and the noise is low. Figure 8b shows the power stabilities of the output pulse laser measured with a power meter (S121C, Thorlabs). Owing to the use of PM fiber, the power stability is only affected by pump power and the fluctuation of the average power is only 0.08% (RMS) in over 6 h.

Fig. 8
figure 8

RF spectrum and power stability of the 72.7-fs compressed pulses, a RF spectrum measured at 1 MHz span with resolution bandwidth of 1 kHz, b power stability measured in over 6 h

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4 Conclusion

In summary, we propose and implement a spectrum-filtering method to optimize the input pulses of a self-similar amplifier. In the experiment, a SESAM mode-locked Yb fiber oscillator is used to generate input pulses for the amplifier. The pulse preshaper consists of a grating pair and a 3.9-nm NBF. We first compress the pulses and then use the NBF to clip the spectrum from 15.5 to 4.0 nm. Meanwhile, the pulse duration is lengthened from 118.2 to 480.9 fs, making its approach to the calculated optimal value ∆Tin, opt. To make the laser system compact, we use one laser diode to pump two Yb fiber sections: the first section is used to pre-amplify the pulses, and self-similar amplification is achieved in the second section. The self-similar amplifier generates linearly chirped parabolic pulses with a spectrum of 42.6 nm, which is nearly three times broader than that of the oscillator output. Using a grating pair, we can dechirp the pulses to 72.7 fs. The dechirped pulses have an average power of 182 mW and pulse energy of 4.9 nJ, which is adequate for biomedical applications such as two-photon microscopy.

In future, the self-similar amplifier described here can be further extended in several directions. First, to choose bandpass filter with appropriate width, we can estimate the ∆Tin, opt by using Eq. (1). However, it should be noted that the spectrum-clipping leads to pulse energy attenuation. Considering that the aim of fiber amplifier is to boost pulse energy, we cannot choose filters with over-narrow width. The choice of the bandpass filter is worth noting. Second, scaling from the experimental results presented here, with LMA Yb fiber, we expect that output pulse energy of a self-similar amplifier will reach higher level after input pulse duration is optimized. In the high-energy self-similar amplification, stimulated Raman scattering induced by long Yb fiber length can disturb linear chirp and limit the achievable pulse energy. The preshaping method we proposed can optimize input pulses to accelerate the parabolic evolution, thus self-similar amplification can be realized in a relatively short Yb fiber. Third, although we use free-space components to verify the preshaping method, the laser system can be made in an environmentally stable all-in-fiber architecture. The grating pair and the NBF in the preshaper can be replaced with a CFBG and a micro-optic fiber component, respectively. We can also clip the spectrum firstly and then compress the pulses. Sub-200 fs pulses will be stretched quickly in silica fiber but longer pulses after spectrum-filtering will be less affected by material dispersion of pigtails of fiber components before the self-similar amplifier. This is another advantage of using the spectrum-filtering method in self-similar amplifiers.