## Appendix A
For the electromagnetic wave frequency with \(\omega>\
u_{ew}\) we obtain, using the approach presented in [43]  
\[\dot{Y}_{e}\ =\ 2\
u_{ew}Y_{vib}\ =\ \
u_{ew}\frac{e^{2}E_{0}^{2}}{2m\omega^{2 }}\ =\ \
u_{ew}\frac{I}{cn_{\text{ecr}}}. \tag{1}\]  
Here, \(Y_{vib}\) is the average kinetic energy of the vibrational motion of electrons in the optical wave with amplitude \(E_{0}\) and frequency \(\omega\), \(n_{\text{ecr}}=m\omega^{2}/(4\pi\epsilon^{2})\) is the critical electron density and \(I=cE_{0}{}^{2}/(8\pi)\) is the wave intensity. In our particular case (the wavelength \(\lambda=1064\) nm), we obtain \(n_{\text{ecr}}\approx 3\times 10^{19}\) cm\({}^{-3}\). The only way to decrease the number of electrons in the process of avalanche development is their adhering to the liquid bubble wall. The frequency of such a process is \((\langle v_{e}\rangle/R_{0})w_{a,e}\), where \(w_{a,e}\) is the probability of electron adhesion. If ionization frequency \(\
u_{i}>(\langle v_{e}\rangle/R_{0})w_{a,e}\), the electron avalanche should start; in accordance with our model, this is the condition of the breakdown threshold. The avalanche development rate depends on the pump intensity \(I\); for large \(I\), an essential ionization can occur at the stage of the inertialretention of the nanobubble, i.e., before its radius substantially grows. The increase in the number of electrons \(N_{e}\) inside a bubston due to the avalanche development is determined by the equation  
\[dN_{e}/dt\ =\ N_{e}/\theta(I), \tag{10}\]  
where the avalanche time constant \(\theta=\theta(I)\) obeys the condition  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]  
Here, the parameter \(I_{0}=cn_{ee}\Delta w_{a,e}\). During the avalanche development, i.e., when an electron loses its energy to ionization, the electron distribution over the energies \(f(Y_{e})\) may be considered to be stationary within the range \(0<\)\(Y_{e}<\Delta\), since the electrons with energy \(Y_{e}>\Delta\) quickly disappear due to the ionization of molecules on the liquid wall. The function \(f(Y_{e})\) can be determined approximately on the basis of the following simple assumptions. The probability \(f(Y_{e})dY_{e}\) is proportional to the duration of electron location in the energy interval (\(Y_{e}\), \(Y_{e}+dY_{e}\)), i.e., is proportional to \(dY_{e}/\dot{Y}_{e}\), where \(\dot{Y}_{e}\) means \"genuine\" (not the average, by contrast to Equation (10)) energy growth rate; i.e., \(\dot{Y}_{e}\) is proportional to the \"genuine\" electron velocity \(v_{e}\ =\ \sqrt{2Y_{e}/m}\). Thus, we obtain \(f(Y_{e})\ =\ c/\sqrt{Y_{e}}\), where \(c\) is a dimensional constant, and after normalizing to the energy interval (0, \(\Delta\)) we arrive at \(f(Y_{e})\ =\ Y_{e}^{-1/2}/2\sqrt{\Delta}\). The mean energy \(\langle Y_{e}\rangle\), which will be called the electron avalanche temperature \(T_{e}\), is equal to  
\[\langle Y_{e}\rangle\ =\ T_{e}\ =\ \int_{0}^{\Delta}Y_{e}f(Y_{e})dY_{e}\ =\ \Delta/3,\]  
and the mean arithmetical velocity is \(\langle v_{e}\rangle\ =\ \sqrt{3T_{e}/2m}\). Surface ionization energy \(\Delta\) can be much less than the ionization energy of an individual molecule of liquid. In the following estimates we will assume \(\Delta\approx 6\) eV (see, e.g., [48; 49]), and, correspondingly, \(T_{e}=2\) eV, \(\langle v_{e}\rangle=7\times 10^{7}\) cm/s, and the frequency of collisions \(\
u_{ew}=7\times 10^{12}\) s\({}^{-1}\). Electron avalanche development inside a nanobubble with the radius \(R_{0}<<l_{em}\) has one more peculiarity as compared with the case \(R_{0}>>l_{em}\). This follows from the fact that here the molecules of the liquid wall, rather than gas molecules, are being ionized, and therefore the avalanche electrons form a \"pure\" electron gas (but not a quasi-neutral plasma), which is confined by the sphere of radius \(R_{0}\). The surface of this sphere obviously acquires a uniformly distributed positive charge \(Q=e\times N_{e}\).