1 Introduction

Laser oscillators with ultrashort pulses and high peak power are demanded tools for applications such as multiphoton microscopy and nonlinear frequency conversion, in particular, supercontinuum generation and ultra-broadband infrared radiation generation. Typically, the repetition rate of an ultrafast oscillator for multiphoton microscopy is about 80 MHz [1]. However, as demonstrated in zebrafish, lower photodamage and higher fluorescence excitation can be achieved with shorter pulses and lower repetition rates at the same average power [2, 3]. With the onset of three-photon imaging, higher peak power and a repetition rate in the range of a few MHz become favorable [4]. The average power required for multiphoton microscopy varies in the range of 5–1000 mW depending on the losses in the microscope and the type of sample. Therefore, compact and stable sources can be provided by Kerr-lens mode-locked (KLM) bulk oscillators with the necessary amount of peak power. In contrast to fiber oscillators, it is easier to scale the peak power of these lasers due to the free choice of the resonator configuration and the beam size. For KLM oscillators already operating at bandwidth-limited pulse durations, the peak power is increased by scaling the pulse energy. For a given average power, the pulse energy can be scaled up by decreasing the pulse repetition rate with an extension of the cavity length [5, 6]. For KLM Ti:sapphire oscillators operating in the negative dispersion regime, this scaling led to pulse energies of 150 nJ, corresponding to 3 MW, at a repetition rate of 6 MHz [7]. However, further peak power increase was limited due to high nonlinearities in the gain medium, which simultaneously served as a Kerr-medium. Thus, in the positive dispersion regime, further energy scaling up to 180 nJ, corresponding to a peak power of 4.5 MW after pulse compression, was performed at 11 MHz [8]. When comparing the reported peak power values [5,6,7,8] to high repetition rate oscillators [9], it becomes evident that the peak power does not scale in the same way as the repetition rate. The increase in peak power was obtained not only in exchange for a low repetition rate but also for a longer pulse duration, even when operating in the negative dispersion regime. This seems to be attributed to the limiting nonlinear effects within the gain medium. The highest intracavity peak power for the low repetition rate oscillators discussed above was about 12 MW [7], which is only approximately twice the value resulting for the high repetition rate oscillator reported in [9].

However, much higher peak powers directly from oscillators became available with the advent of mode-locked Yb:YAG thin-disk oscillators [10]. In the case of Kerr-lens mode-locking [11], the intracavity peak power is typically well above 100 MW and can reach values of 1 GW [12, 13]. The recent development led to a record peak power outside the oscillator of 100 MW at about 15 MHz [14,15,16]. In KLM Yb:YAG thin-disk oscillators, the Kerr-lens is separated from the gain medium, in which the oscillator mode can reach diameters of several mm, removing the limiting nonlinear effects at this place. For mode-locking, the Kerr-lens is then produced by a separate Kerr-medium, which is placed in the beam caustic of a 4f-telescope cavity extension. In applications with lower power requirements, bulk lasers can offer a compact and less complex alternative.

For KLM bulk lasers, the separation of Kerr- and gain medium originally appeared for the purpose of pumping a Nd:YLF gain medium with diode lasers [17]. The pump spot of the low brightness radiation required a correspondingly larger laser mode diameter inside the laser gain medium, which prohibited the onset of sufficient Kerr-lens strength. Therefore, the laser mode was focused into a separate Kerr-medium within the cavity. Furthermore, a highly nonlinear separate Kerr-medium has been employed to lower the pump power threshold for KLM in a Ti:sapphire oscillator [18]. For KLM Ytterbium bulk oscillators, the separation of Kerr- and gain medium again offered convenient pumping with large pump spots from powerful diode lasers [19,20,21,22,23,24]. This development led to average powers reaching 10 W in KLM operation of a Yb:CALYO oscillator [24]. The use of an external Kerr-medium made it possible to choose from different nonlinear materials to optimize the peak power. Thus, peak powers up to 1.85 MW, corresponding to about 12 MW intracavity, at 50 MHz have been reported for an Yb:CALYO bulk oscillator [23]. Similar to KLM thin-disk lasers, the main contribution of the Kerr-lens is provided by a separate Kerr-medium, which allows for a flexible adjustment of the Kerr-lens strength and the nonlinear phase shift without being limited by the mode diameter inside the gain medium and its nonlinear properties [11, 12, 25]. This flexibility is particularly relevant if the resonator’s repetition rate is decreased to either scale up the peak power [12] or to adapt the repetition rate to MHz levels.

Here, peak power scaling for fixed average power and pulse duration is demonstrated with an Yb:YAG bulk oscillator containing a separate Kerr-medium. The focus lies here on the demonstration of the general principle of the peak power scaling by cavity extensions and adjustments of the Kerr-lens and nonlinear phase shift, which can be conveniently accomplished with the help of a separate Kerr-medium. However, we think that this type of peak power scaling is generally transferable to bulk oscillators with other gain media, for which, in part, the approach with a separate gain medium was already adopted, e.g., Yb:CALGO [26], Yb:CALYO [23], Yb:KGW [22, 27], and Yb:Lu2O3 [20]. The future development of these oscillators might be of interest for nonlinear microscopy. Additionally, Cr:ZnS oscillators in the 2 µm spectral range might benefit from this approach [28, 29], which are currently increasing in relevance for ultra-broadband infrared generation [30, 31].

2 Experimental setup

Here, we systematically study the peak power scaling behavior by increasing the pulse energy throughout four oscillator configurations. We start with the high repetition rate configuration with 70 MHz and thus low peak power and go toward an almost seven times increased cavity length by inserting 4f-telescope extensions based on spherical mirrors at the position of the flat output coupler mirror (Fig. 1). The beam caustic of our cavity configuration is shown in Fig. 2. Importantly, the cavity is designed in such a way that the beam diameter inside the gain medium is approximately two to three times larger than the beam diameter inside the Kerr-medium. Moreover, the repetition rate can be lowered without significantly changing the oscillator beam caustic inside the cavity due to the imaging condition of the telescope extensions [5]. This is a crucial consideration, as Kerr-lens mode-locking is highly sensitive to the oscillator mode diameter inside gain and Kerr-medium.

Fig. 1
figure 1

Schematic of the KLM Yb:YAG bulk oscillator without cavity extensions for a repetition rate of 70 MHz. OC: flat output coupler mirror (wedged), IC input coupler mirror, DM dispersive mirror, HR high reflective mirror, R1/R2 concave mirror (ROC = 300 mm), R3/R4 concave mirror (ROC = 100 mm), Yb:YAG gain medium, KM Kerr-medium, and HA circular hard aperture

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Fig. 2
figure 2

Calculated beam caustic for the empty cavity in the 70 MHz configuration, which is the basis for all other configurations, at the center of the stability region. The approximate placement of the gain medium (GM) and the Kerr-medium (KM) is marked by arrows. All longer cavity configurations are realized by 4f-telescope extensions added on the right end. The beam caustics for the longer configurations are, therefore, unaltered within the 70 MHz base cavity due to the imaging condition of the extensions. Inset: Zoomed in at the foci of the curved mirror pairs

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The gain medium is a 2 mm thick Yb:YAG crystal with an ytterbium doping concentration of 5 at. %. The front and back surfaces are antireflection (AR) coated for perpendicular incidence of pump and oscillator modes. The cavity is based on a standard X-cavity, whereby the gain medium is placed at the center position between two identical concave spherical mirrors, R1 and R2, with a radius of curvature (ROC) of 300 mm. These mirrors are selected to achieve a minimum beam waist diameter for the oscillator mode of ca. 100–120 µm. With this enlarged oscillator mode diameter inside the gain medium, we drastically decrease Kerr-lensing inside the gain medium. Mode-locking just with the Kerr-lens from the gain medium was, therefore, not possible. A separate Kerr-medium was introduced to provide the necessary Kerr-lens. It is placed inside the cavity between a second set of identical concave spherical mirrors R3 and R4 with a ROC of 100 mm. Here, a minimum oscillator mode diameter of ca. 40–60 µm is generated. To consider the actual impact of the Kerr-lens on the resulting self-amplitude modulation (SAM) and, therefore, on the mode-locking mechanism, it would require a more detailed analysis, which also involves the actual position of the Kerr-lens with regard to the focus position of the cavity mode [22, 32,33,34,35]. As the involved parameters are difficult to assess and for simplicity, we are just referring to the Kerr-lens strength to obtain an indicator for the relevance of the Kerr-lens from the gain medium. The Kerr-lens strength decreases with the increasing beam diameter, i.e., decreasing peak intensity inside the gain medium. The following formula gives an estimation for the behavior of the Kerr-lens strength 1/f under the assumption of a thin plate of matter [36, 37]:

$$\frac{1}{f} = \frac{{8n_{2} lP_{{\text{p}}} }}{{\pi w^{4} }} ,$$
(1)

with a nonlinear refractive index n2, a Kerr-medium thickness l, a peak power Pp, and a mode diameter w. Considering the thickness of the gain medium of 2 mm and a Rayleigh length of about 10 mm, this assumption seems acceptable. The strong dependence on the mode diameter allows us to clearly distinguish the strength of the Kerr-lens inside the gain medium from its strength inside the Kerr-medium [20]. Subtle adjustment of the Kerr-lens strength and the nonlinear phase shift is possible by shifting the Kerr-medium along the beam around the focus generated by the surrounding set of concave mirrors. In a coarse way, the Kerr-lens is adjusted by the selection of the material itself, which has a specific nonlinear refractive index, as well as by the selection of the material thickness. The materials ZnS with a thickness of 3 mm and SrTiO3 with a thickness of 2 mm are used owing to their extraordinarily high nonlinear refractive indices of ca. 9 and \(5 \times 10^{ - 19}\) m2/W, respectively [38]. For pumping the gain medium, we use a commercial fiber-coupled diode laser emitting a power of up to 10 W at a wavelength of around 940 nm. The fiber has a core diameter of 105 µm and a numerical aperture of 0.15. We image the fiber tip into the gain medium in a 1:1 configuration with two aspherical lenses with a focal length of 50 mm each [20]. The enlarged mode diameter inside the gain medium ensures a suitable overlap of pump and oscillator modes. However, the exact laser mode diameter in the gain medium is dependent on the exact resonator alignment suitable for mode-locking and the Kerr-lens itself. The given value is, therefore, just a rough estimation. To access the gain medium with the pump radiation, we had to fold the cavity close to the gain medium with a flat input coupler (IC) mirror. A wedged output coupler (OC) mirror with a reflectivity of 95% was chosen to ensure a sufficient amount of intracavity peak power to provide reproducible mode-locking during all the experiments. Negative dispersion is introduced by mirrors providing a flat group delay dispersion (GDD) of – 2400 fs2 per bounce over a bandwidth of 6 nm. For finer adjustment, mirrors with a GDD of – 400 fs2 per bounce are introduced. Adding a circular hard-aperture at the other end of the cavity provides the necessary SAM for mode-locking. The right balance is found by carefully scanning the distance between the concave mirrors R3 and R4, as well as the Kerr-medium position. Finally, mode-locking is initiated by slightly knocking the OC mirror.

3 Results and discussion

The 70 MHz configuration delivers pulses with a peak power of up to 93 kW (Table 1). Decreasing the repetition rate in three steps down to 10.7 MHz by different 4f-telescope extensions of the cavity leads to an increase of the peak power up to 424 kW. The scaling behavior appears to be linear with regard to the cavity length, respectively the cavity roundtrip time (Fig. 3), while the average power and the pulse duration are kept almost constant at 600–700 mW and 90 – 100 fs, respectively (Table 1). Hence, we could demonstrate peak power scaling, which is attributed only to the increase of pulse energy due to cavity length extensions. However, during the scaling, the average power dropped about 100 mW.

Table 1 Parameters for KLM Yb:YAG bulk oscillators with different pulse repetition rates
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Fig. 3
figure 3

The peak power shows a linear scaling behavior when the length, i.e., the roundtrip time of the cavity, is increased. The inset shows the beam profile at 10.7 MHz

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Fig. 4
figure 4

The spectra of all four resonator configurations were fitted around the center region under the assumption of a sech2-shape. The integration over the fit and the measured spectrum reveals that up to 17% of the power is contained in the Kelly sidebands for the 10.7 MHz configuration (Table 1)

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To ensure still stable soliton pulses for the increased pulse energy values but unchanged pulse duration, we have to adjust the nonlinear phase shift, i.e., the self-phase modulation (SPM) coefficient, and the dispersion of the cavity. According to the soliton area theorem, the full width at half maximum (FWHM) pulse duration τ is maintained by keeping a balance between dispersion and SPM [39]:

$$\tau = 1.76\frac{{\left| {D_{{{\text{RT}}}} } \right|}}{{\gamma_{{{\text{SPM}}}} E_{{\text{p}}} }} ,$$
(2)

with the roundtrip group delay dispersion \({D}_{\mathrm{RT}}\), the SPM coefficient \({\gamma }_{\mathrm{SPM}}\), and the pulse energy \({E}_{\mathrm{p}}\). Throughout all configurations, the nonlinear phase shift could be flexibly adjusted with the external Kerr-medium. The Kerr-medium is, therefore, shifted around the focus of the laser mode. At the same time, this leads to a readjustment of the Kerr-lens strength and, therefore, to a readjustment of the SAM. Additionally, for the two configurations with the highest peak power, we changed the Kerr-medium from 3 mm thick ZnS to 2 mm thick SrTiO3 to reduce the nonlinear refractive index by about a factor of two (Table 1). Therefore, we needed to adjust the roundtrip dispersion introduced by the mirrors in the range from – 7600 to – 12,400 fs2. Otherwise, mode-locking was observed to break up into multipulsing behavior. Single-pulse operation was ensured by a photodiode with a bandwidth of 350 MHz (Fig. 5) and two autocorrelators with scanning ranges of 15 ps and 3.3 ns, respectively (see Fig. 6).

Fig. 5
figure 5

Pulse train at a repetition rate of 10.7 MHz

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Fig. 6
figure 6

The pulse durations for all four resonator configurations were obtained by fitting the measured intensity autocorrelation (AC) traces under the assumption of sech2-pulses

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However, in the autocorrelation trace, a small elevation appears at a delay of approximately 0.5 ps (Fig. 6). A side-pulse might have appeared due to nonideal dispersion compensation. The smallest step of GDD we could introduce was – 400 fs2. It might be that the GDD was simply not perfectly adjusted to the average power and repetition rate. Higher-order dispersion was not compensated. We did not consider higher-order compensation because of the still relatively long pulse durations. The realized pulse durations were around 90–100 fs for all demonstrated configurations. With a FWHM spectral bandwidth of 11.7 nm (Fig. 5d), a time-bandwidth product of 0.32 follows, which implies a close to Fourier transform-limited pulse duration. In some of our oscillator configurations, even shorter pulse durations of about 70–80 fs could be realized by reducing the dispersion and fine-tuning the nonlinear phase shift. However, the oscillator became increasingly unstable and more challenging to mode-lock. Therefore, we increased the pulse duration to the reported values to ensure stable operation. The mode-locking was then reproducibly initiated by knocking the OC mirror.

In the spectrum, strong Kelly sidebands appear (Fig. 4). The sidebands increase in number and power with increasing peak power throughout the studied configurations. A higher nonlinear phase shift due to the scaling of the intracavity peak power and perturbations in the pulse propagation inside the cavity might lead to their appearance [40]. The use of our highly dispersive mirrors might also contribute here to the formation of the Kelly sidebands, as the GDD quickly changes outside the specified bandwidth of about 6 nm. The presented values of pulse energies and peak powers are corrected based on a fit of the spectra assuming sech2-shape (Table 1) The corresponding correction factor is given by the ratio of the integrals of the fit and the measured spectrum. Even with this excessive nonlinearity, mode-locking was running in a steady and stable operation throughout all cavity configurations, thanks to the cavity configuration providing enough SAM to support it.

Further peak power scaling was limited by the available space to extend the cavity length beyond the 10.7 MHz configuration. Attempts to increase, on the other hand, the average power by more powerful pump diodes led to limiting thermal effects. As the demonstrated peak power scaling does not deviate so far from the expected linear behavior (Fig. 3), we expect that further extensions of the cavity should drive up the peak power even more. There is still enough room for the adjustment of the Kerr-effect by adjusting the position of the Kerr-medium or replacing it with a less nonlinear material. Materials such as YAG or CaF2 should be applicable for intracavity peak powers beyond the current value of 8.5 MW while still keeping the Kerr-lens well above that in the gain medium due to a different oscillator mode diameter there [20, 23]. Assuming a minimum mode diameter of 40 µm inside the Kerr-medium, the intracavity peak power would need to be increased by a factor of several hundred to bring the Kerr-lens strength inside the gain medium to the current level inside the Kerr-medium. Even though the mode diameter inside the Kerr-medium might be larger due to adjustments of its position, it indicates that the current configuration might still be far from being limited by the nonlinear effects inside the gain medium. Therefore, we think that higher intra-cavity peak powers could still be handled. Ultimately, the Kerr lens in the gain medium will become strong and comparable to the Kerr-lens in the separate Kerr-medium. At these peak power levels, our separation and scaling concept will not work anymore. Eventually, at this point, it should become possible to mode-lock the laser without a separate Kerr-medium, only with the Kerr-lens induced in the gain medium. In order to perform further peak power scaling, switching to the thin-disk concept probably makes sense. A compact solution to further increase the cavity length could be realized by inserting a Herriott cell as a delay line into the cavity [5, 6, 41, 42]. Further fine-tuning of the spatial overlap of pump and oscillator modes in the gain medium for soft-aperture mode-locking may increase overall efficiency.

4 Conclusion

In summary, we developed a KLM Yb:YAG bulk oscillator and scaled up the peak power from 93 to 424 kW by extending the cavity length and, therefore, increasing the available pulse energy while keeping the average power at 600–700 mW and the pulse duration around 100 fs. The pulse repetition rate decreased from 70 to 10.7 MHz. Inside the cavity, the peak power increased correspondingly from 1.9 to 8.5 MW. Therefore, the Kerr-lens and the nonlinear phase shift were adjusted for each cavity configuration with a separate Kerr-medium. Further peak power scaling might be possible by an additional cavity length extension by an intracavity Herriott cell. A transfer of the demonstrated peak power scaling to other KLM oscillators could result in compact high peak power sources with MHz level repetition rates for multiphoton microscopy applications.