## 2 Methods and Materials
There are in general two most practical irradiation cases, when (i) filaments do not overlap each other with interspacing \(s_{1}\) several times greater than their characteristic diameter at the 1/2-level of VED decrease, \(2r_{1/2}\), that is:  
\[s_{1}>>2r_{1/2}, \tag{2}\]  
as shown in Figure 1, or (ii) are written inside the sample with a partial overlap of each other, that is, with a smaller spacing, comparable with a filament diameter, \(s_{1}\approx 2r_{1/2}\). Only the case (2) with no overlapping is considered in the modelling below. As can be easily shown, for the assumed Gaussian distribution of \(w\) the \(r_{1/2}\) radius relates with the previously introduced \(r_{f}=r_{1/e}\) as \(r_{1/2}=(\text{ln}2)^{1/2}r_{f}\cong 0.83r_{f}\). The model follows up a concept of an indirect action of laser radiation on the material [40; 41] that has been successfully applied to the deep (high-aspect-ratio) drilling of opaque materials. That is, the incident laser energy is at first transformed into the plasma energy (in the given case, the filament plasma) with a conversion coefficient (absorptivity) \(A_{L\text{-P}}\). This is followed by a conversion of the plasma energy into a modification of the material due to the radial transfer and dissipation of the plasma energy from the filament axis over a certain cylindrical volume around the filament. Such an impact zone is formed due to thermal conductivity and/or shock wave. The conversion coefficient of the plasma energy into the released energy is \(A_{P\text{-}H}\). This indirect laser action concept is universal in the sense that it does not depend on whether the material is transparent or opaque. The thermalisation of the laser-excited electrons, that is, the equalisation of their temperature with the temperature of atoms and ions happens on the timescale of about 1 ps [23; 42; 43]. For example, the relaxation time for electron-hole plasma in fused silica is \(\tau_{e\text{-}h}\) = 0.15 ps [21; 42], and in borosilicate glass BK7 \(\tau_{e\text{-}h}\) = 2 ps [43]. This is comparable with the pulse duration (a few times smaller or greater). That is, an equilibrium temperature maximum of the filament, or at least a relatively small part of it, although great enough in absolute value (few thousands K or more), may be reached for ps and even sub-ps laser pulses already before the pulse termination. Moreover, the filament cools down on a microsecond time scale, and such cooling times are much longer than the mentioned typical time of setting thermal equilibrium. The material's exposure to the high laser flux required for cleaving can induce its temperature to rise from room temperature to thousands and even hundreds of thousands of degrees (see below). This drives the material through a chain of transitions, which are not only a softening or melting, but also a dissociation and ionisation, whereas the electronic structure experiences noticeable rebuild even at the pulse timescale, up to the collapse of the bandgap [44; 45], or at least to such a decrease of the bandgap \(E_{g}\) from its initial typical for transparent materials value 4-9 eV at room temperature to the value less than laser photon energy \(h\
u\) (for example, for the considered below laser wavelengths \(\lambda\) = 1.04 and 0.52 \(\upmu\)m, \(h\
u\) equals to 1.19 and 2.38 eV respectively). Besides, at a high laser excitation, the emission of free-electron plasma created by the laser occurs in a wide spectral range, often with short wavelengths in the UV range, where the material has high linear absorption (10\({}^{4}\)-10\({}^{5}\) cm\({}^{-1}\) or more). The typical duration of this secondary emission is about 1 \(\upmu\)s to tens of \(\upmu\)s, corresponding to the abovementioned times for filament cooling down [43]. This duration is much longer than that one of the ps-, or sub-ps- laser pulse.