## 1 Introduction  
Supercontinuum generation (SCG) plays a pivotal role in many applications such as white-light interferometry [1], optical coherence tomography [2], frequency comb spectroscopy [3], and dual-comb Fourier transform spectroscopy [4]. Moreover, SCG is the backbone of the optical parametric amplification (OPA). Supercontinuum (SC) pulses are used as seed pulses that are to be amplified in the OPA systems by high energy pump pulses. This nonlinear interaction creates the possibility of generating longer wavelength pulses through the generation of the idler pulses [5; 6; 7; 8]. Using OPA systems, the realm of few-cycle pulse generation has fallen into our grasp [9; 10; 11; 12; 13; 14], and even attosecond pulse generation systems operating based on OPA have been established [15; 16; 17]. Unlike optical parametric generation (OPG) sources, in which the carrier-envelope phase (CEP) is uncorrelated with the CEP of the pump, SC sources inherit the CEP of the applied pump, and if the same source were used to pump both the SCG and OPA, passive CEP stability could be achieved for the idler pulses as a result. One of the most commonly-used SCG methods, specially for OPA applications, is to focus intense laser pulses into a bulk material [18]. SCG in bulk has been used in OPA systems as one of the most prominent seed sources [19; 20; 21; 22; 23; 24], because supercontinua generated in bulk media generally exhibit a very well-defined phase and good beam quality, which make them ideal seed sources for OPA systems. Alternatively, ultra-broadband SC can be achieved using optical fibers [25]. The largest bandwidth observed for SC generated in fibers comes at the cost of introducing considerable instability and spectral modulation, as we showed in our previous work in [26], which may render them inapplicable for OPA applications. Similarly, intrinsic properties of the self-phase modulation (SPM) that manifest in the form of complicated temporal pulse splitting and a modulated spectral profile limit the applicable medium length for the SCG in bulk. For this reason, the spectral broadening achieved by the visible and NIR sources in bulk media has not achieved similar bandwidth boundaries as those of the optical fibers [27; 28; 29]. As a result, such SCG systems lack the longer wavelength components necessary for the mid-infrared (MIR) applications such as MIR-OPA. To increase the bandwidth of SC, it is important to study the evolution of the nonlinear effects within the materials. Previous studies have shown that the dispersion of the medium can greatly affect the dynamics of the self-focusing [30]. It has been shown that the self-focusing comes to an end by the pulse splitting effect for SCG systems operating at the normal dispersion regimes [31]. Pumping SCG systems at anomalous regions, on the other hand, would result in the extension of the length of the filament and ultra-broadband achievable spectrum [18; 32]. Considerable extension of the IR cut-off limits has been reported in previous literature [33; 34; 35; 36; 37]. The choice of the medium for SCG depends on a number of factors, among which the most important ones include the damage threshold, transparency range, the nonlinear refractive index, and the required cut-off limits. Sapphire is considered to be an ideal material pumped by NIR sources such as Ti:sapphire lasers, which would cover the 450-1600-nm range [38; 39; 40]. Using calcium fluoride (CaF\({}_{2}\)) crystals can help extend the limits of the generated SC into the UV down to 250 nm. The application of CaF\({}_{2}\) comes with the downside of being prone to photo-damage [41], which makes them difficult to maintain. Yttrium aluminum garnet (YAG) has been shown to be one of the best candidates for MIR-SCG, because of its low band-gap and high damage threshold [42; 43]. The broad bandwidth of the SC is the result of multiple phenomena, including self-focusing, self-phase modulation, pulse splitting, four-wave mixing, self-steepening, etc. [44; 45; 46; 47].