Solitary beam propagation in periodic layered Kerr media enables high-efficiency pulse compression and mode self-cleaning

Generating intense ultrashort pulses with high-quality spatial modes is crucial for ultrafast and strong-field science and can be achieved by nonlinear supercontinuum generation (SCG) and pulse compression. In this work, we propose that the generation of quasi-stationary solitons in periodic layered Kerr media can greatly enhance the nonlinear light-matter interaction and fundamentally improve the performance of SCG and pulse compression in condensed media. With both experimental and theoretical studies, we successfully identify these solitary modes and reveal their unified condition for stability. Space-time coupling is shown to strongly influence the stability of solitons, leading to variations in the spectral, spatial and temporal profiles of femtosecond pulses. Taking advantage of the unique characteristics of these solitary modes, we first demonstrate single-stage SCG and the compression of femtosecond pulses from 170 to 22 fs with an efficiency >85%. The high spatiotemporal quality of the compressed pulses is further confirmed by high-harmonic generation. We also provide evidence of efficient mode self-cleaning, which suggests rich spatiotemporal self-organization of the laser beams in a nonlinear resonator. This work offers a route towards highly efficient, simple, stable and highly flexible SCG and pulse compression solutions for state-of-the-art ytterbium laser technology.

times larger, which could lead to compact device footprint. Nevertheless, the propagation of strong femtosecond pulses in a bulk medium is more challenging, also because of the high nonlinearity. The self-focusing effect can result in catastrophic beam collapse and optical damage, when the power is higher than the critical power Pcr [8]. Recently, the progress in SCG with multiple thin plates of Kerr media has attracted great attention [9][10][11][12][13][14]. By placing the beam self-focusing in the free space between material plates, this technique elaborately circumvents the optical breakdown, allowing substantial enhancement of nonlinear Kerr interaction [10,12].
Although generation of few-cycle pulses has been demonstrated with a two-stage compressor [13], the existing implementations of multiplate SCG and pulse compression have several limitations, because the beam propagation is still not well controlled. First, complicated space-time coupling leads to strong conical emission, which can cause energy loss >40% [12].
Second, because of the space-time-coupling-induced higher-order dispersion, the dispersion compensation requires custom-designed chirped mirrors or pulse shapers in some cases [12,13], and strong pedestals can be occasionally observed in the compressed pulses [13]. Finally, the plate thickness and their positions have been empirically determined for the broadest spectrum, making it difficult to be systematically studied and repeated.
Here, we want to point out that the optimum condition for SCG in layers of Kerr media is to form discrete solitons, which can sustainably propagate and interact with the medium under high intensity. Discrete solitons are localized wave-packets propagating with a stable structure in nonlinear periodic structures, thanks to the balance between diffraction and material nonlinearity [8,15]. Fundamentally, understanding the formation of solitons is important, because they represent stable solutions for the cubic nonlinear Schrödinger equations 4 (NLSE) [16,17], which is related to many branches of physics [18,19]. Theory has predicted that it is possible to have solitary spatial modes in periodic layered Kerr media (PLKM) [20][21][22]. Experimentally, repetitive variation of the beam profiles for P≈6 Pcr has been observed without spatial beam collapse [23]. Ultimately, such periodic propagation of femtosecond pulses can be considered as two-dimensional transverse [(2+1)D] discrete spatial solitons, with a repetitive spatial mode on the material layers [15,21]. The stability of these solitons, however, could be strongly influenced by space-time coupling during the propagation. To date, no experimental studies on the influence of spatiotemporal propagation exist. Neither have any shown how to identify these solitary modes, or studied their potential applications for SCG and pulse compression.
In this Letter, we experimentally characterize the space-time coupled propagation of femtosecond pulses with a peak power reaching GW in a nonlinear resonator comprising PLKM, and successfully identify the formation of discrete solitary modes under a range of conditions. By comparing the experimental and theoretical results, we reveal a universal relationship between the characteristic beam size and the critical nonlinear phase of the solitary modes, under which the propagation of light wave-packets is localized in both space and time. We also show that the formation and breakdown of solitons is manifested by the correlated variations in spectral, spatial and temporal profiles of femtosecond pulses. In practice, we demonstrate two applications of these solitary modes -First, taking advantage of the localized solitary modes, we can successfully suppress the spatial loss and higher-order dispersion, and achieve ~8-fold pulse compression with ~90% efficiency in a single-stage compressor. This result demonstrates the unique advantages of the solitary modes over the empirically implemented multiplate 5 SCG [9,10,13]. Second, spatial mode-self-cleaning with high efficiency is demonstrated when the beam propagation is on resonance, which can be attributed to the spatial self-organizing effect in a nonlinear resonator.
The repetitive propagation of a high-intensity laser beam in PLKM can be regarded as a cavity resonator with intensity-dependent non-spherical (Kerr-lens) mirrors (see Fig. 1(a)).
Because of the complexity of the spatiotemporal effects introduced by Kerr nonlinearity [8], the NLSE simulation relies heavily on numerical analysis, which complicates understanding of the fundamental physical processes. Here, we first resort to the Fresnel-Kirchhoff diffraction (FKD) integral to identify the self-consistent solitary modes. Assuming the normalized amplitude of the incident optical field as U1, we derive that the amplitude, after propagating through a unit of the resonator and right before the next period, is given by where ρ and ρ' are the radial coordinates rescaled by L  and J0 the zeroth-order Bessel function. Here, one period of the resonator contains a layer of Kerr medium with a thickness of l and the following layer of free space by length L. In Eq. (1), b represents the nonlinear phase given by 20 , where I0 is the field intensity. The Fox-Li iteration is then used to numerically find the solitary modes [24], the nonlinear phase associated with which is defined as the critical nonlinear phase bc. As shown in Fig. 1(b), the characteristic properties of the solitary modes are determined by the beam radius (w) on the layers, the resonator length (L) and the critical nonlinear phase (bc). Here, we define a Fresnel-number-like beam radius squared 2 w L  , for convenience of discussion.
In our experiments, we employed a Yb:KGW amplifier laser system with a pulse duration imaging setup, while the temporal intensity profiles are measured by second-harmonicgeneration frequency-resolved optical gating (SHG-FROG) [25].
First, we present findings on the key features of the resonator stability and soliton formation. In the experiments, we observe a reduction of the far-field beam size and an improvement of the spatial mode quality (see SM), as the incident pulse energy (Ein) approaches from low to a critical value under a specific resonator length L (e.g. L=50.8 mm in the inset of Fig. 1(b)). When the pulse energy is low and the self-focusing in the media is weak, the beam propagation is dominated by diffraction, leading to a diverging beam size on the successive layers ("unstable diffractive region" in Fig. 1(b)). On the contrary, when the incident beam reaches the critical nonlinear phase (bc) and produces sufficient self-focusing to appropriately balance the diffraction (Ein=260 μJ for L=50.8 mm, corresponding to a field intensity of 5.0×10 12 W/cm 2 ), the laser beam can repetitively propagate through the PLKM with a wellconfined beam size. Indeed, with the 4-f imaging measurements, we observe that the beam radius approaches a stable value on the material layers after the self-adjustment in the first few layers under the resonant condition ( Fig. 1(c)), which indicates the formation of discrete spatial 7 solitons [21]. We summarize the experimental results under different Ls in Fig. 1(b), which exhibits excellent agreement with the FKD model. We note that the excellent agreement between our results and the FKD model indicates that the formation of spatial solitons is fundamentally caused by the balance between the diffraction and nonlinear self-focusing effects.
Meanwhile, since the temporal profiles are not considered in the FKD model, such agreement also suggests that the femtosecond pulses should have a stable temporal structure under the spatial soliton modes throughout the propagation. This is confirmed by the direct measurement of the temporal profiles, as shown in Fig. 1(d). As a result, we can define these solitary modes as temporally confined (2+1)D spatial solitons.
As shown in the inset of Fig. 1b [22]. The geometrical factor γ is given by L l  = . However, we find that the width of this region is generally narrower in the experiments than the FKD results. This could be attributed to the temporal pulse splitting above the resonance (see below), which is not considered in the FKD model. Beyond 2Pthr, the strong Kerr lens breaks the balance between the self-focusing and diffraction, and the beam size grows out of limit within few periods of propagation ("unstable dissipative region", Fig. 1b).
In the regions beyond the resonance, the space-time coupling breaks the solitary modes and strongly influences the spectral, spatial and temporal profiles (see Fig. 1a). First, we find that the bandwidth ceases to increase almost immediately when b>bc, as shown in Fig. 2a. 8 Correspondingly, we observe temporal pulse splitting and asymmetric pulse profiles (Fig. 2c), accompanied by the strong enhancement of conical emission. These effects can be understood as the correlated effects resulting from the spatial cavity-mode selection for different pulse intensities in time. When the peak intensity is higher than the critical value, the pulse temporal center mismatches with the cavity resonance and experiences fast divergence due to the strong Kerr lensing. This breaks the solitary modes and results in splitting of pulses in time and conical emission ( Fig. 1(a)). This explains the widely observed strong conical emission in many previous experiments [10,12,13]. The saturation of spectral bandwidth, on the other hand, is caused by the SPM process on a temporally split and positively chirped pulse [26]. These correlated effects can be observed for different resonator lengths, indicating the universality of these effects. Meanwhile, they can be well captured by the 2D NLSE simulations with an accuracy of Ein within 10% of the experimental values (see SM).
In Fig. 2(d) and (e), we plot the time-frequency analysis of the output pulses under different conditions (see SM). In the dissipative region, significant amount of nonlinear chirp can be observed in the long-wavelength side of the spectrum (Fig. 2(e)). In stark contrast, majority of the optical energy is linearly chirped under the solitary mode ( Fig. 2(d)). We note that, since the pulses have passed through the same amount of Kerr medium in both cases, we can exclude that such difference is caused by the higher-order dispersion of materials or carried by the input pulses. On the other hand, when the femtosecond pulses are split and asymmetric in time, nonlinear frequency chirp can be produced by the space-time coupling and SPM process [27].
Indeed, as we have shown in Fig. 1(d), the complex pulse profiles have already been generated through few material layers in the dissipative region. The higher-order dispersion here has 9 significant effects on pulse compression. Even with an appropriate compensation of negative GDD, strong pedestals spanning ~100 fs in time can still be clearly observed for the compressed pulse in Fig. 2(e), which contributes to ~24% of the overall pulse energy (SM). In contrast, the pedestals can be well suppressed when the pulse propagation is under the solitary mode (see Fig. 3(a)).
The generation of GW, temporally confined spatial solitons in this work can significantly improve the SCG and pulse compression. Here, we summarize three advantages. First of all, as evidenced in Fig. 1(c) and Fig. 1(d pulses from this single-stage compressor is close to the TL pulse (see Fig. 3(a)). The experimental and reconstructed FROG traces are shown in Fig. 3(b) and (c), respectively. We note that our result represents the greatest number of layers implemented in a single-stage multiplate SCG experiment, and the SCG bandwidth is about 50% broader than the previous work under the similar conditions [13]. Secondly, the single-peaked temporal profile under the solitary modes (Fig. 2(f)) also avoids generating higher-order dispersion, which circumvents the usage of custom-designed chirped mirrors or pulse shapers [12,13]. This is evidenced by the fact that the clean pulse compression (Fig. 3(a)) is achieved by only compensating the second-order dispersion. Last but foremost, the spatial loss induced by conical emission is strongly suppressed by the solitary propagation. As shown in Fig. 3(d), the conical radiation only contributes <10% of the total output energy when the propagation is on resonance, and this contribution can increase to ~35% in the dissipative region (Fig. 3(e)). Overall, by combining the broad SCG spectrum and suppressed loss from space and time, we achieve ~5 times increment in the peak power in a single-stage compressor, as shown in Fig. 3(a), which represents the largest increment of pulse peak power from a single-stage compressor with multiple layers of Kerr media (see SM).
Finally, we show the spatial mode self-cleaning from the nonlinear PLKM resonator. The improvement of the output spatial mode, in terms of the circularity and the intensity profile, can already be observed after the fundamental laser beam passing through the PLKM (see SM).
To further investigate the mode-self-cleaning effect, we introduce substantial perturbance on the beam profile, by inserting a cylindrical beam blocker with a diameter of 0.8mm into the laser beam, which has a full-width-of-half-maximum (FWHM) size of ~3 mm ( Fig. 4(a)). The spatially modulated laser beam is then focused into a resonator with bc=0.5, L=25.4 mm, consisting of 20 layers. As shown in Fig. 4(b-d), by matching with the cavity resonance, the output mode is significantly cleaned and transforms to the resonator solitary mode. In Fig. 4(e), we plot the filtering efficiency as a function of the blocker transverse position Δx, with larger Δx inducing greater spatial modulation ( Fig. 4(a)). The PLKM resonator can support a high filtering efficiency (>85%) across a large range of spatial modulation, in direct contrast to an ideal linear spatial filter (see SM). This result suggests that the solitary modes here represents attractors of the (2+1)D cubic NLSE equation [18]. There must be an efficient pathway, in 11 which the laser energy in the higher-order spatial modes can be transferred to the solitary modes through the repetitive Kerr interactions. This is consistent with the spatial self-organization of laser beams, previously observed under the nonlinear interactions in filamentation [28], selffocusing collapse [29] and multimode fibers [30,31].
In summary, we experimentally investigate the spatiotemporal propagation of strong femtosecond pulses in PLKM resonators and reveal its influence on the stability of optical solitons. Taking advantages of the unique characters of these solitary modes, we demonstrate high-efficiency SCG, pulse compression and spatial mode-self-cleaning. These results are relevant to a wide range of applications, such as Kerr-lens mode locking [32], ultrashort-pulse generation [10,12,13] and high-energy wavelength scaling [11]. Moreover, the results here illustrate the general features of the space-time coupling of solitons under periodically modulated Kerr nonlinearity, which may reinforce the theory and help understanding soliton formation under similar periodic "potentials" in many other fields of nonlinear optics, including waveguide arrays [33,34], periodic refractive-index gratings [35,36] and photonic crystal fibers [37], as well as in condensed matter physics [38,39] and in biology [40].