## 3 Model  
### Typical Durations of the Processes of Filament Creation and Relaxation of Its Energy  
LFC includes the processes of laser excitation and relaxation of the material, whose characteristic durations may differ by orders of magnitude [22]. We will clarify them for our modelling conditions, and also introduce some new time parameters that have not been previously noted. As mentioned above, the time \(t_{eq}\) for equalising the temperature of laser-excited electrons with the temperature of atoms and ions is of the order of 1 ps, and this is comparable to the pulse duration \(\tau\), which for LFC is, as we will see below, of the order of 0.15 \(\div\) 2 ps, that is \(\tau\)-\(t_{eq}\), or \(\tau<t_{eq}\), or \(\tau>t_{eq}\). The time of an inertial confinement of pressure in a heated filament \(t_{P1}\) and the time of pressure confinement in the space between neighbour filaments \(t_{P2}\) can be estimated as the time of passage of a sound wave with a speed \(c_{s}\) of the filament diameter \(2r_{f}\) and the distance between neighbour filaments \(s_{1}\):  
\[t_{P1}\approx 2r_{f}\ /c_{s}\,t_{P2}=s_{1}/c_{s}, \tag{3}\]  
Until these times have elapsed, it can be assumed that the shock wave and any related destruction do not propagate into the surrounding material from a highly heated filament or from the space between neighbour filaments. When substituting into Equation (3) the characteristic value of the sound velocity in the material \(c_{s}\approx\) 6 \(\times\) 10\({}^{5}\) cm/s [47], the submicron radius of the filaments used in this method \(r_{f}\) = 0.25-0.5 \(\upmu\)m and, for example, \(s_{1}\)\(\approx\) 2.5-5 \(\upmu\)m, one gets the estimates: \(t_{P1}\approx\) 0.08-0.17 ns and \(t_{P2}\approx\) 0.4-0.8 ns. As seen, \(t_{eq}\)\(<<t_{P1}\), \(t_{P2}\). The so-called \"heat confinement time\" in the filament is estimated as [39]:  
\[t_{T1}=r_{f}^{2}/(4\chi). \tag{4}\]  
When the radiation energy is deposited into the filament plasma and then is released in the filament in the form of its heating for a time smaller than \(t_{T1}\), it can be assumed according to the heat conduction theory that there is practically no radial heat loss beyond the filament radius. Also, with a proper selection of the bulk energy density of the filament \(w\), as we will see below, there is a practical absence of plasma energy losses due to thermal radiation in the transparency region of the surrounding material. For example, for a submicron filament radius \(r_{f}\) = 0.25-0.5 \(\upmu\)m in a fused silica glass, when \(\chi\) =5 \(\times\) 10\({}^{-3}\) cm\({}^{2}\)/s, one has \(t_{T1}\) = 30-125 ns. For sapphire, when \(\chi\) = (1.5-2.4) \(\times\) 10\({}^{-2}\) cm\({}^{2}\)/c, one gets \(t_{T1}\) = 6-40 ns. It is seen that \(t_{P1}\), \(t_{P2}\)\(<<t_{T1}\). The time of displacement of the laser beam relative to the material by a distance equal to the filament radius \(r_{f}\) is:  
\[t_{r}=r_{f}/(s_{1}f). \tag{5}\]  
By analogy with the pressure and the heat confinement times \(t_{P1}\), \(t_{P2}\) and \(t_{T1}\), the parameter \(t_{r}\) can be called as a beam position retaining time. Here \(u=s_{1}f\) is the speed of relative motion of the beam and the material surface. Irradiation of the material with one or several pulses for a time much less than \(t_{r}\) corresponds to the localisation of the energy deposition of these pulses in a space region with a radius that practically does not differ from the filament radius \(r_{f}\). For the submicron radius \(r_{f}\) = 0.25-0.5 \(\upmu\)m for tempered glass, when \(u=s_{1}f\) =100-400 cm/s, we get \(t_{r}\) = 125 \(\div\) 500 ns, and for sapphire, when \(f\) = 3 MHz, \(r_{f}\) = 0.5 \(\upmu\)m, \(s_{1}\) = 4 \(\upmu\)m, and \(u=s_{1}f\) =11 m/s, we get \(t_{r}\) = 45 ns (the \(u\) values indicated here see in the Section 4 \"Discussion\" below). It is seen that \(t_{T1}<<t_{r}\), or also can be \(t_{T1}\approx t_{r}\).