1. Introduction

Harmonic mode-locking operation enables high repetition rates from long resonators with easy set-up. It is well known from fiber laser technology [1, 2], in which harmonic frequencies are typically imposed by intra-cavity phase [3] or amplitude [4] modulation or passively by colliding-pulse harmonic mode-locking [5]. Solid-state lasers have been harmonically mode-locked with intra-cavity phase modulation of a Nd:YAG rod laser [6] or passively with diode-pumped and SESAM mode-locked laser system [7, 8], or by using an etalon [9] or the coupled-cavity effect [10].

In this work, a SESAM mode-locked diode-pumped Yb:CaGdAlO₄ (Yb:CALGO) bulk oscillator producing femtosecond pulses at a fundamental repetition rate of f_fund = 94 MHz is presented. By increasing the pump power, second harmonic mode-locking operation with a repetition rate of f_2nd−harm = 188 MHz and third harmonic at f_3rd−harm = 282 MHz is observed. For this operation no means of active stabilization is required. A suppression of the fundamental radio frequency signal at second harmonic operation with more than 88 dBc and at third harmonic with more than 71 dBc verifies the quality of this operation regime. Long-term passive stability was observed for over 160 h. In order to trace the multi-pulse dynamics, the pulse evolution from pulse splitting up to the second harmonic stable operation was tracked with a fast photo diode and a fast real-time oscilloscope. In order to understand the high quality of passive harmonic mode-locking, a numerical model is presented which describes the divergence of two initially closely-spaced pulses based on the gain evolution. The presented results indicate the mechanisms of harmonic mode-locking and compose a concept for bulk material based systems, which are further scalable in power and repetition rate.

2. Experimental setup

The oscillator, shown in Fig. 1, is based on an a-cut Yb:CALGO crystal of 1.5 mm thickness and a Yb doping of 5 %. Yb:CALGO has a broad and smooth absorption cross section with a maximum at 979 nm [11–13]. It provides a broadband emission plateau for the laser regime between 1000 and 1050 nm [12–14], which would allow for pulse durations down to 32 fs [15–17].

 

Fig. 1 Schematic diagram of the experimental setup of the Yb:CALGO laser oscillator. The laser crystal is pumped through a dichroic mirror. The cavity is formed by HR mirrors, GTI mirrors for dispersion compensation, a sapphire plate in Brewster’s angle, the SESAM as one end mirror, and an output coupler with 2.4 % transmission. The radius of curvature for the curve mirrors are HR_2,ROC = −200 mm, HR_4,ROC = −300 mm, HR_5,ROC = −400 mm and HR_6,ROC = −600 mm.

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A fiber-coupled 30 W diode laser with a 200 µm core diameter (JENOPTIK, JOLD-30-FC-12-980) is imaged 1:1 into the 138 µm resonator beam waist (1/e² radius) inside the crystal for pumping. For an optimized absorption of pump light the emitted wavelength of the laser diode was shifted to 979 nm by controlling the operating temperature. A polarized beam splitter and a motorized half-wave plate were used to variably control the pump power. A second lambda-half plate was used to optimize the pump absorption. The cavity was set-up by highly reflective (HR) and Gires-Tournois interferometer (GTI) dispersive mirrors. To induce mode-locking, a semiconductor saturable absorber mirror (SESAM, BATOP GmbH, SAM-1040-1-500fs) was installed at one cavity end. It is specified with a high reflectivity from 1020 nm to 1100 nm, a zero group delay dispersion (GDD) from 1040 nm to 1070 nm, a modulation depth of 0.6 %, non-saturable losses of 0.4 %, a saturation fluence of 120 µJ/cm², and a relaxation time constant of about 500 fs. As second end mirror a 2.4 % output coupler is employed. We operate the oscillator in the solitary regime at negative GDD. The GTI mirrors can be chosen for a net GDD of D₂ = −420 fs², −1470 fs² or −2680 fs² per round trip. A sapphire plate in Brewster’s angle is used to operate the oscillator in p-polarization. The total resonator length of 3.2 m provides a fundamental repetition rate of f_rep = 94 MHz. Depending of the intracavity GDD, pulse durations at full width at half maximum (FWHM) from τ_FWHM = 605 fs down to τ_FWHM = 190 fs have been observed.

3. Experimental results

For increasing pump powers at the different GDD values, pulse duration, optical spectrum, and the radio frequency (RF) spectra have been recorded. The red crosses in Figure 2(a)–2(c) show the pulse durations as a function of the intra-cavity average power for the three different dispersion regimes. The label I indicates the single, II the double, and III the triple pulsing regime, respectively. At approximately P_int,−2680 = 132 W, P_int,−1470 = 80 W and P_int,−420 = 37 W of intracavity average power, indicated by red lines in the figures, the single pulse splits into two pulses, and the pulse duration jumps to maintain the solitary condition. Beyond around P_int,−1470 = 130 W and P_int,−420 = 64 W three pulses oscillate. This solitonic pulse splitting is well-known as an effect stemming from two-photon absorption in the SESAM [18–20]. Beyond P_int,−420 = 70 W Q-switching led to damage on the SESAM. Table 1 summarizes the average output power P_out, the pulse durations τ_FWHM, the pulse energies E_out, and the output peak powers P_max,out for the individual pulses at different GDD D₂. Going to harmonic orders beyond three was limited by the pump power in the high dispersion regime and by SESAM damage at −420 fs². For a short time only, fourth harmonic mode-locking at −630 fs², with an average power of 3.26 W, 240 fs pulse duration and a sideband suppression better than 75 dBc was observed; however, this regime was quite sensitive to the focal position on the SESAM and is therefore disregarded in the following.

 

Fig. 2 Pulse duration (red crosses) and radio frequency measurements (color map) for net intracavity dispersion D₂ of (a) −2680 fs², (b) −1470 fs² and (c) −420 fs² as a function of the internal average power.

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Table 1. Measured output parameters at different harmonic mode-locking (HML) orders (single-pulse operation at −420 fs² with a cw component in the spectrum).

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After pulse splitting, stabilization of the multiple pulses at the harmonic distances was observed. The color maps in Figure 2(a)–2(c) visualize radio frequency spectra for the three different dispersion values. The extinction of certain harmonic lines in the RF spectrum indicates high-fidelity harmonic mode-locking. Figure 3(a) reveals 88 dB of suppression in 1 Hz resolution bandwidth; Fig. 3(b) shows the third harmonic with 71 dB suppression.

 

Fig. 3 (a) Damping of the fundamental repetition rate peaks in the second harmonic and (b) the third harmonic mode-locking regime (−1470 fs²) with P_{int,2nd−harm} = 107 W and P_{int,3rd−harm} = 174 W of intracavity average power.

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The long-term passive stability of the harmonic mode-locking operation can be demonstrated by tracking the suppressed second harmonic peak in the RF signal with a 1 Hz resolution. This is shown in Fig. 4 for more than 160 hours of laser operation.

 

Fig. 4 Long-term tracking of the suppressed second harmonic peak around 188 MHz in third harmonic operation mode.

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The pulse splitting was accompanied by a systematic shifting of the optical spectra in agreement with Zhou et al. [8].

In Figure 5(a), the optical spectra (color map) and the center frequencies (red dots) follow the resonator internal average power in the three mode-locking regimes. This is directly attributed to the solitary pulsing condition [21–23]

 

Fig. 5 (a) Measured optical spectra (color map) and central frequency (red crosses) as a function of resonator internal average power for a net dispersion of −1470 fs². (b) Measured optical spectra for 51 W (red), 83 W (green), 100 W (blue), 135 W (purple) and 144 W (black) internal power.

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In order to understand the dynamics of the split pulses we tracked the temporal evolution with a fast photo diode and a real-time oscilloscope with 16 GHz bandwidth and 50 GS/s sampling rate (Tektronix DPO71604C). It was used in the "FastFrame"-mode with a dead time of 200 µs between the measured frames. Figure 6 shows a typical pulse splitting event and the slow temporal separation of both pulses until after about four seconds the stable position at half of the round trip time (red dashed line) is reached. The pulses stabilize their temporal position according to the second harmonic of the repetition rate without any active control. Particularly interesting is the observation of certain temporal plateaus during the transient. They correspond to the effective optical path lengths between specific resonator components: The lowest plateau indicates the collision of two pulses on mirror GTI₁ (see Fig. 1). The second plateau is attributed to the collision in the crystal, and the third to mirror HR₅. On these components, the two pulses collide and interact, e.g., via nonlinear coupling in the gain medium [24] and/or scattering at the mirrors. All surfaces inside the resonator might give rise for building a plateau, and other measured transients show other plateaus. It is unclear why certain components surfaces stabilize the pulse positions during one transient, during another not.

 

Fig. 6 Pulse splitting and evolution of the second harmonic mode-locking regime. The distances of the four plateaus to the first pulse at time 0 ns are GTI₁ ≈ 0.94 ns, crystal ≈ 3.72 ns, HR₅ ≈ 4.99 ns and the red dashed line indicates half the round trip time T_R/2 ≈ 5.33 ns.

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4. Theoretical model

In accordance to Zhou et al. [8], the final harmonic arrangement of the pulses is attributed to the gain dynamics. To understand the relative motion of two pulses in the resonator we adapted a theoretical model based on the Master equation of mode-locking and the split-step Fourier method [25–29]:

Figure 7 reveals the evolution of the gain (red) and the total losses (green) together with the pulse (blue) in the double pulse regime. Here, for clarification of the effect, the repetition time was reduced to T_R = 1 ns. Further parameters are: E_sat,L = 6.8 mJ, $n_2=9\cdot {10}^{-20}\frac{{\mathrm{m}}^2}{\mathrm{W}}$, Aeff = 5.7 · 10⁻⁸ m², λ = 1050 nm, τ_L = 380 µs, ${\mathrm{\Omega }}_{\mathrm{g}}=2.1\cdot {10}^{13}\frac{1}{\mathrm{s}}$, E_sat,A = 0.88 nJ, D₂ = 550 fs², τ_A = 850 fs, q₀ = 1 %, and l = 0.5 %. The higher gain for the leading pulse results in a–slightly higher gain gradient giving rise to a very slow relative pulse motion away from each other. At the distance of half the round trip, the gain and the gradient become symmetrical and the pulses stabilize (see Fig. 6).

 

Fig. 7 Evolution of gain (red) and losses (green) during two pulses (blue).

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5. Conclusion

A harmonically mode-locked Yb:CALGO laser oscillator is presented, featuring harmonics of the repetition rate up to the third order, limited only by pump power. Harmonic operation is demonstrated with exceptionally high fidelity and long-term stability for more than 160 h. The transient evolution of the pulse positions and the convergence against the steady harmonic state have been experimentally tracked, and the dynamics have been explained by a gain evolution model. This is easily transferable also to other gain media. Based on this technique, high power, high-repetition rate oscillators with an simply implementable set-up long resonator are available.