## 3 Model
### Typical Durations of the Processes of Filament Creation and Relaxation of Its Energy
The effective time \(t_{E}\) for the release of absorbed radiation energy in the atomic subsystem when it relaxes to a temperature equilibrated state is estimated in order of magnitude by one or another from two parameters \(t_{e\text{-}h}\) and \(t_{e\text{-}i}\) depending on laser pumping:  
\[t_{E}=t_{e\text{-}h}\text{ or }t_{E}=t_{e\text{-}i}. \tag{6}\]  
Here \(t_{e\text{-}h}\) is a characteristic relaxation time of bound electrons in the electron-hole plasma, which is obtained at relatively low fluences or VED, i.e., electrons located at the electronic levels of the material excited by laser photons [21]. Numerically \(t_{e\text{-}h}\) ranges from values close to \(t_{\alpha}\) (i.e., from 0.15-2 ps mentioned above) to few tens of picoseconds, when the formation of laser-induced self-trapped excitons and related defects is taken into account [47,48]. Formation of other long-living states can also contribute. Another parameter here, \(t_{e\text{-}i}\), is an effective relaxation time of free charge carriers of electron-ion plasma obtained at a relatively high VED. It corresponds to the time of transfer of the excitation energy of ions and free electrons of the plasma to heat by their recombination and de-excitation both in the plasma itself and on the sidewalls of the filament. The source of excitation for such plasma can be both laser photons and photons of plasma thermal emission. Their energy can be quite large. We remind readers here that, according to the Wien displacement law, the wavelength of the maximum of the thermal emission spectrum of a black body varies with the temperature \(T\) as \(\lambda_{max}\) [nm] = 257/\(T\) [eV]. The photon energy corresponding to this wavelength is \(h\
u_{max}\) [eV] = 1240/\(\lambda_{max}\) [nm] = 4.83 \(T\) [eV]. As follows from the shape of the broad spectrum of black body thermal emission, photons with the twice greater energy \(h\
u^{*}\)[eV] = 2\(h\
u_{max}\) = 9.7 \(T\) [eV] give although smaller, but a quite comparable contribution to the thermal emission spectrum with respect to the photons \(h\
u_{max}\) in the maximum. For example, the substitution into these expressions of a fairly moderate plasma temperature \(T\) = 5 eV gives \(\lambda_{max}\) = 51 nm, \(h\
u_{max}\) = 24 eV and \(h\
u^{*}\) = 48 eV. The maximum \(\lambda_{max}\) appears to be located in the region \(\lambda\leq\lambda_{ha}\) of strong absorption (>10\({}^{4}\)-10\({}^{5}\) cm\({}^{-1}\)) of the cold material surrounding the filament, where \(\lambda_{ha}\) is a boundary of this high absorption region. For display glass \(\lambda_{ha}\) = 218 nm and for fused silica--140 nm. As seen, the plasma with the indicated temperature provides the presence of thermal emission photons of the high energy of 24-48 eV, which are effectively absorbed by the material. These energies are a few tens of times higher than the laser photon energy, which is 1.19 eV for laser wavelength \(\lambda\) = 1040 nm, or 2.38 eV for \(\lambda\) = 520 nm. The value of \(t_{e\text{-}i}\) is not well characterised quantitatively in the literature. The LFC with high VEDs considered in this paper is accompanied by dissociation and disordering of the material. For example, for silica glass, such dissociation starts at a temperature about 2500 K and more [49], which, as we will see below, is relatively low compared to temperatures of few tens of thousands K, that can be reached in LFC with high VEDs. It looks reasonable to expect in such case an intensive formation of defects in the material structure that can make a significant contribution to the formation of long-living electronic states. They may be important for our simulation if an energy relaxation and release through such states could provide a significant contribution to the overall balance of released energy. However this subject has not been well studied yet.