2.1 B-integral management with discrete SCF configuration

Since obtaining high average power femtosecond pulses at a high repetition rate is inevitably accompanied by serious thermal effects in the bulk amplifiers, gain medium doping with a lower concentration of active ions but longer gain region is a straightforward solution. However, the B-integral value (B) will increase in this case due to the continuous accumulation of nonlinear phase shift in the long gain medium (L)^[ Reference Koechner^43]:

(1) $$\begin{align}B=\frac{2\pi }{\lambda }{\int}_0^L{n}_2(z)I(z)\mathrm{d}z,\end{align}$$

where n ₀ and n ₂ are the linear and nonlinear refractive indexes, respectively, λ is the wavelength and I is the beam intensity.

Then, the laser beam starts self-focusing as the peak power exceeds the critical power of P _cr = 3.77λ ²/(8πn ₀ n ₂)^[ Reference Lu, Tsou, Chen, Chen, Cheng, Yang, Chen, Hsu and Kung^44], thus leading to a reduction of the achievable gain, degraded beam quality and even breakup of the amplified pulses. Although the controllable beam waists of the free-space propagation light in the SCF can mitigate nonlinear effects more effectively than those in conventional silica fibers, the laser performance is still adversely affected when amplifying ultrashort pulses at a high-power level, as demonstrated in the Yb-SCF femtosecond laser amplifier^[ Reference Zhao, Zheng, Huang, Gao, Dong, Tian, Yang, Chen and Petrov^40].

To overcome this limitation, the strategy employed here is based on a discrete layout of the SCFs, which can provide the following advantages: (i) prevention of continuous accumulation of nonlinear phase shift, that is, discrete B-integral values, and (ii) the length of the SCF is controllable and can be shorter than twice the self-focusing length (Z _f), assuming the average power amplifier reaches P _max at the middle of the last-stage SCF amplifier^[ Reference Koechner^43]:

(2) $$\begin{align}L<2{Z}_{\mathrm{f}}=\frac{2{\pi \omega}_0^2}{\lambda }{\left(\frac{8\pi {n}_0{n}_2{P}_{\mathrm{max}}}{3{\lambda}^2}-1\right)}^{-1/2},\end{align}$$

where ω ₀ is the beam waist radius.

This ensures that the laser pulses exit the SCF before beam collapse, analogous to the multiple-plate structure^[ Reference Lu, Tsou, Chen, Chen, Cheng, Yang, Chen, Hsu and Kung^44], but with a much lower light intensity in our case. With such a discrete configuration of SCFs (three identical SCFs with 50 mm in length, see the next section for more details), the simulated average power can achieve more than 200 W for 500-fs pulses at a high repetition rate of 50 MHz, with the B-integral value of less than π and 2Z _f of approximately 50 mm.

2.2 Experimental setup of the seed source

To perform the direct amplification experiment, Figure 1 schematically shows the entire SCF amplifier system comprising a femtosecond seed source and pre- and multi-stage SCF amplifiers without temporal pulse stretching or compression. The seed source was a wavelength-tunable semiconductor saturable absorber mirror (SESAM) mode-locked Tm(2.8%, atomic fraction):LuScO₃ ceramic (3 mm × 3 mm × 3 mm) laser in-band pumped by a home-made continuous-wave (CW) erbium-doped yttrium aluminum garnet (Er:YAG) bulk laser delivering the maximum power of 3.2 W, with a linearly polarized beam at 1645 nm^[ Reference Zhang, Wang, Chen, Wang, Liu, Zhao, Jia, Zhang and Sun^45]. The pre-amplifier was a Ho:YAG SCF pumped by a laser diode (LD) at 1907 nm, and the main amplifier based on three discrete Ho:YAG SCFs was in-band pumped by a Tm-fiber laser at 1907 nm.

Figure 1 Schematic diagram of the Ho:YAG SCF amplification system. L1/L2, optical collimation/focusing system; M1, M2, M3, concave dichroic mirrors; CM1, CM2, plane dispersive mirrors; OC, output coupler; BF, birefringent filter; DM, plane dichroic mirror; LD, laser diode.

For the seed laser, as shown in Figure 1, the collimated pump beam was focused into the ceramic sample by an aspheric lens (L2, f = 75 mm), resulting in a beam diameter of approximately 45 μm. A standard X-shaped astigmatically compensated cavity with a physical length of approximately 2.1 m was employed, in which two plano-concave mirrors, M1 and M2, both with a radius of curvature of R _OC = −100 mm, formed a beam waist of 30 μm (sagittal) × 60 μm (tangential) diameter within the Tm:LuScO₃ ceramic placed at the Brewster’s angle between them. Another plano-concave mirror (M3) with R _OC = −150 mm was used to create a second beam waist of 80 μm in diameter on the GaSb-based SESAM^[ Reference Zhang, Song, Zhou, Liu, Liu, Zhang, Xu, Xue, Xu, Chen, Zhao, Griebner and Petrov^46] for mode-locking. Two chirped mirrors, CM1 and CM2, providing negative group delay dispersion (GDD) of −125 and −1000 fs² per bounce, respectively, were inserted into the other cavity arm for dispersion compensation. This arm was terminated by the output coupler (OC), a plane-wedged mirror with a transmission of T _OC = 3%. A home-made 3-mm-thick quartz birefringent filter (BF), cut with its crystal axis at 24° to the surface normal, was inserted close to the OC at Brewster’s angle to tune the wavelength of the femtosecond laser for matching the wavelength of the subsequent SCF amplifiers.

2.3 Experimental setup of the direct femtosecond amplification system

After passing through an optical isolator (ISO), the seed pulse was directly imaged into the pre-amplifier via two lenses (f = 200 mm). A 1-mm-diameter (Φ), 50-mm-long Y₃Al₅O₁₂ (YAG) SCF doped with 0.3% (atomic fraction) Ho³⁺ ions served as a pre-amplifier counter-pumped by a 30 W fiber-coupled LD (400-μm core diameter, 0.22 numerical aperture (NA)) at 1907 nm. The pump light was imaged into the initial part (~3 mm from the pump face) of the SCF using a 1:1 telescope system and then waveguided down to the fiber end through total internal reflection. The pre-amplified femtosecond pulses were directly imaged into the power amplifier, where three identical Ho:YAG SCFs (Φ 1 mm × 50 mm, 0.5% Ho³⁺) in a serial arrangement with separations of approximately 4 mm were used as gain media. The beam waist with a measured diameter of approximately 460 μm was located in the initial part of the second SCF to ensure free-space propagation of the laser beam in the three-stage SCF power amplifier. A high beam quality (M ² ~ 1.05) Tm-fiber laser at 1907 nm was employed as a pump source to ensure spatial matching with the beam from the pre-amplifier along the entire SCF power amplifier chain. The pump light was focused into the second SCF to a spot diameter of approximately 450 μm, with the beam size in the three SCFs not exceeding 600 μm in diameter at any position, thus lengthening the effective gain regions. This makes the experimental setup extremely simple and robust without requiring any re-imaging telescopes. It shall be outlined that all lateral surfaces of the SCFs have been optically polished, and the measured transmission losses were less than 3 dBm⁻¹ at 632.8 nm. All input/output SCF faces and all lenses were antireflection-coated. To better mitigate the thermal effects, all the SCFs were mounted in specially designed aluminum modules being in direct contact with cooling water at 13°C.

3.1 Seed source and pre-amplifier systems

The output performance of the mode-locked Tm:LuScO₃ laser is shown in Figure 2. For a pump power of approximately 2.5 W, the average output power ranged from 0.24 to 0.35 W when adjusting the intra-cavity BF, corresponding to a tuning spectral span of approximately 63 nm ranging from 2044 to 2107 nm, as shown in Figure 2(a). The autocorrelation traces in Figure 2(b) were recorded by a commercial autocorrelator (APE pulseCheckSM, extended IR), yielding pulse durations (τ) of 195–373 fs.

Figure 2 Output performance of the Tm:LuScO₃ laser. (a) Tunable mode-locked operation: spectra and average output power (circles and dashed line) and (b) the corresponding autocorrelation traces where τ is the pulse duration (full width at half maximum, FWHM) assuming sech²-temporal shapes. (c) Optical spectrum, (d) autocorrelation trace with the inset showing the recorded near-field spatial beam profile, (e) radio frequency (RF) spectrum of the fundamental beat note measured with a resolution bandwidth (RBW) of 100 kHz and (f) RF spectrum on a 1 GHz span (RBW = 300 kHz) at the selected seed wavelength of 2091.6 nm. SNR, signal-to-noise ratio.

To match the peak wavelength of the Ho:YAG emission spectrum at approximately 2091 nm, the central wavelength of the seed laser was tuned to 2091.6 nm with a spectral full width at half maximum (FWHM) of approximately 12 nm; see Figure 2(c). The average output power at this wavelength reached 0.45 W at a pump power of 3.2 W, corresponding to a single pulse energy of approximately 6 nJ. The measured autocorrelation trace is presented in Figure 2(d), and the corresponding pulse duration (FWHM intensity) was 360 fs.

Figures 2(e) and 2(f) depict the radio frequency (RF) spectrum of the mode-locked Tm:LuScO₃ seed laser at 2091.6 nm. The fundamental beat note at approximately 75.45 MHz exhibited an SNR of more than 65 dBc. The uniform harmonic beat notes observed on a 1 GHz span indicate the absence of spurious modulations and the stable mode-locked operation. The beam quality factor at the maximum average power was M ² = 1.01 and 1.04 in the horizontal (x) and vertical (y) directions, respectively. The inset in Figure 2(d) shows the near-field beam intensity profile, which shows the TEM₀₀ mode of the seed laser. The excellent temporal and spatial features are prerequisites for subsequent amplification in the Ho:YAG SCFs.

The Ho:YAG SCF pre-amplifier was designed to increase the average power of the seed laser, and the maximum output power of 0.8 W was reached for an injected seed power of 0.45 W. The amplified pulse spectrum was centered at 2091.5 nm with an FWHM of 10.5 nm (Figure 3(a)), which corresponded to slightly longer pulses of 410 fs (as shown in Figure 3(b)) attributed to the gain narrowing, as well as being slightly affected by material dispersion (see Section 1 in the Supplementary Material for the discussion). The M ² values measured after the pre-amplifier were only marginally increased, M_x ² = 1.11 and M_y ² = 1.15 in the horizontal and vertical directions, respectively. Figure 3(d) shows the pump intensity distribution in the pre-amplifier under wave-guiding conditions, which was simulated by the ray-tracing method. The pump beam was focused into the rear part of the Ho:YAG SCF with a spot diameter of approximately 400 μm, which was well matched to the beam size of the seed laser, ranging from approximately 380 to 420 μm in the SCF. Following the initial focusing, the pump beam diverged and then was confined through total internal reflection, thus generating multiple pump gain regions along the SCF. The transverse spatial intensity distribution maintained azimuthal uniformity at different locations along the SCF, as shown in Figure 3(d), which provides favorable conditions for femtosecond pulse amplification compared with traditional rod amplifiers.

Figure 3 Performance of the Ho:YAG SCF pre-amplifier. (a) Optical spectrum and (b) autocorrelation trace measured after the pre-amplifier. (c) Beam caustics and near-field intensity profile (inset) with fitted M ² propagation factors. (d) Simulated pump light spatial intensity distribution in the Ho:YAG SCF of the pre-amplifier pumped by the 1907 nm LD with Z indicating the distance from the input pump face.

Figure 4 Input–output characteristics of the power amplifier. (a) Measured average output power, (b) net gain and (c) optical extraction efficiency versus absorbed pump power for different input levels; η _slope, slope efficiency with respect to absorbed pump power. (d) M ²-factors of the multi-stage Ho:YAG SCF power amplifier versus absorbed pump power measured at P _in = 0.8 W; the insets show near-field output beam profiles recorded at absorbed pump power levels of 10 W (left) and 90 W (right) with x and y designating the horizontal and vertical directions, respectively.

3.2 Power scaling in the multi-stage discrete Ho:YAG SCF amplifier concept

The input–output characteristics of the multi-stage Ho:YAG SCF power amplifier were first evaluated by inserting a variable attenuator into the output beam path of the pre-amplifier, enabling continuous adjustment of the injected power without affecting other beam characteristics. The performance of the power amplifier for input average powers P _in of 0.1, 0.5 and 0.8 W at a constant pulse duration of approximately 410 fs is shown in Figure 4. At P _in = 0.1 W, the maximum average output level from the power amplifier reached P _out = 17.8 W at approximately 75.45 MHz with an incident pump power of 106 W, corresponding to a slope efficiency of η _slope = 34.2% with respect to the absorbed pump power, and a gain value of approximately 178. Such a remarkably high gain in the single-pass amplifier is attributed to the high-brightness pumping and the effective mode-matching in the multi-stage SCF configuration. The relatively low output power and slope efficiency at P _in = 0.1 W can be attributed to the low extraction capacity at low seed power levels. For P _in = 0.8 W and the same incident pump level, P _out reached 56.3 W with a slope efficiency of 60.4%, corresponding to a gain value of more than 70. Under these conditions, the residual pump power behind the multi-stage SCF power amplifier amounted to only approximately 5%, which leads to an optical extraction efficiency η _opt = (P _out − P _in) / P _abs of 55.8%. To the best of our knowledge, this is the highest average power demonstrated from a 2-μm femtosecond solid-state amplifier. The absence of roll-off in the power dependence suggests that further scaling should be possible by increasing the pump power. However, the high gain of the Ho:YAG SCF (particularly at the peak emission wavelength of ~2090 nm) induces the generation of self-oscillation when the pump power is further increased, thus limiting the power scaling. This will require optimization of the SCF design, for example, enhancing signal transmittance of the end facet, tilting the device at a slight angle or cutting the end face at a small angle, which will disrupt the conditions of the self-oscillation, and further power scaling is expected. Moreover, the power scaling performance by employing single and double Ho:YAG SCFs as power amplifiers is detailed in Figures S1 and S3 in the Supplementary Material.

Figure 5 Spectral and temporal performance of the Ho:YAG SCF power amplifier. (a) Spectra and (b) autocorrelation traces measured at the output of the power amplifier chain at the maximum pump level for the three different seed average powers; τ is the derived pulse duration (FWHM intensity). The gray spectrum in (a) and the right-hand scale show the stimulated emission cross-section σ_SE of Ho:YAG. Dynamic evolution of (c) the optical spectrum and (d) the autocorrelation trace from the seed laser through the pre- and power amplifier at the maximum seed level.

Moreover, the self-focusing phenomenon is most likely to occur in the final-stage SCF gain medium theoretically owing to its elevated amplification power (~56.3 W), potentially leading to an increased B-integral value. However, based on the self-focusing length (Z _f) and self-focusing critical power (P _cr) formulas presented in the main text, even although the pulse peak power is close to P _cr (~2.1 MW), the calculated Z _f ≈ 80 mm substantially exceeds the 50-mm SCF length used experimentally. Consequently, the self-focusing phenomenon can be neglected in the discrete SCF configuration, yielding a total B-integral of approximately 0.75. For a 150-mm SCF, Z _f is significantly shorter than the gain medium length, enabling the self-focusing process to occur within the crystal. Under this condition, the calculated total B-integral is approximately 8.3, which exceeds the value under the discrete layout condition by almost a factor of 10.

For the maximum P _in = 0.8 W, the output beam quality factors at different absorbed pump powers were measured with a Spiricon charge-coupled device (CCD) camera, as shown in Figure 4(d). The beam quality factors remained below 1.2 over the entire measurement range, which confirms the spatial mode stability of the amplified beam. At an absorbed pump power of approximately 90 W, the beam quality factor was measured to be M_x ² = 1.18 and M_y ² = 1.14 in the horizontal and vertical directions, respectively. It can be concluded that the beam quality of the pre-amplifier is practically maintained in the power amplifier, while thermal stress-induced degradation is suppressed by direct water cooling.

3.3 Dynamic evolution of the optical spectra and temporal properties

The output spectra from the power amplifier under maximum pump power were similar for the three injected power levels of 0.1, 0.5 and 0.8 W, corresponding to pulse durations of 778, 859 and 866 fs, respectively; see Figures 5(a) and 5(b). To analyze the spectral and temporal evolution, the measured spectra and pulse durations are summarized in Figures 5(c) and 5(d) for all stages (including those of the mode-locked laser and the pre-amplifier). Compared with the spectrum of the seed laser centered at approximately 2091.6 nm, the FWHM of the spectra gradually decreased, and the central wavelength shifted to approximately 2090.6 nm. Meanwhile, a weaker spectral peak appeared at approximately 2096 nm, which correlates with the gain spectrum of Ho:YAG (see Figure 5(a)). The spectrum of the power amplifier at the average power of approximately 56 W shows no indication of amplified spontaneous emission (ASE) or nonlinear effects, which highlights the main advantages of the SCF, namely, high overall gain and nonlinear effects suppression. To overcome the gain narrowing in the power amplifier, an effective solution in future work would be initial spectral shaping with suppressed peak intensity of the seed light^[ Reference Malevich, Andriukaitis, Flöry, Verhoef, Fernández, Ališauskas, Pugžlys, Baltuška, Tan, Chua and Phua^15, Reference Malevich, Kanai, Hoogland, Holzwarth, Baltuška and Pugžlys^47].

The autocorrelation trace at maximum output power of the power amplifier indicates a pulse duration of 778 fs without any significant pulse distortion. Remarkably, the ultrashort pulse was amplified up to 0.75 μJ pulse energy and 0.84 MW peak power without a stretcher–compressor configuration. Such high performance of the ultrafast amplifier (i.e., high efficiency, high average power and good beam quality) obviously benefited from the long gain region, the nonlinear effects suppression and the large surface-to-volume ratio of the multi-stage Ho:YAG SCF amplification system. Similarly, the dynamic evolution of the optical spectra and temporal properties are shown in Figures S2 and S4 in the Supplementary Material.