## Appendix A  
If a gas nanobubble (bubston) exists in the focal area of a laser beam, an electron avalanche can develop inside this nanobubble. The process of an avalanche developing depends on the relationship between the nanobubble (bubston) radius \(R_{0}\) and the average free path of electrons \(l_{cm}\) with respect to their collisions with gas molecules inside the bubston. The length \(l_{cm}=1/(n\sigma_{em})\) where \(\sigma_{em}\) is the transport cross-section of electrons scattering on the molecules, \(\sigma_{em}=(2-4)\times 10^{-16}\) cm\({}^{-2}\), \(n=p/T\) is the density of the molecules and \(p\) is the pressure on the bubble surface (hereinafter \(T\) is expressed in energy units). For the bubston, the surface tension pressure \(2\alpha/R_{0}\) (\(\alpha\) is the surface tension coefficient) is compensated by the electrostatic expansive (negative) pressure \(Q_{0}{}^{2}/(8\pi\varepsilon R_{0}{}^{4})=2\pi\sigma^{2}/\varepsilon\), caused by the adsorption of ions on the inner bubble surface (\(\sigma\) is the surface charge density and \(\varepsilon\) is the dielectric constant) [29]. This is why, for bubstons, pressure \(p\) is always equal to hydrostatic pressure \(p_{0}\), which is close (under laboratory conditions) to the external (atmospheric) pressure at the liquid interface. Accordingly, under normal conditions \(n=n_{L}=2.7\times 10^{19}\) cm\({}^{-3}\) (the Loschmidt number). Thus, the length \(l_{cm}\) is about \(10^{-4}\) cm and always exceeds the bubston radius \(R_{0}=100\) nm, see [30]; in this work, the radii of individual bubstons were found in the experiments with dynamic light scattering and phase microscopy. The opposite inequality \(l_{cm}<R_{0}\) can take place under normal conditions only for unstable bubbles, which can exist under non-stationary conditions. For such bubbles, the pressure \(2\alpha/R_{0}\) is not compensated, and therefore \(p=p_{0}+2\alpha/R_{0}>p_{0}\). As is known, such bubbles are not stable; see, for example, [44]. However, for water and some other liquids under normal conditions, the difference between \(p\) and \(p_{0}\) is valid only for \(R_{0}<10^{-4}\) cm. This is why for \(R_{0}>10^{-4}\) cm we have \(p=p_{0}\), \(n=n_{L}\) and, accordingly, \(l_{cm}=10^{-4}\) cm \(<R_{0}\). It will be shown below that such bubbles can be generated at the final stage of the stimulated optic coalescence (SOC) of bubston clusters. The SOC effect is the basis for the suggested model of optical breakdown in liquids. Qualitatively, this model is as follows. The development of an electron avalanche inside bubstons, which compose a cluster, leads to the rise of pressure inside individual bubstons, resulting in their expansion, and to the merging into a \"large\" bubble. This is the SOC effect. As we have already mentioned, bubston clusters consist of stable gas nanobubbles (bubstons) with radii \(R_{0}<<l_{em}\). Initial free electrons (i.e., the electrons that always exist inside the nanobubble) arise due to the adsorption of anions at the interface of inner liquid walls, see [31]. Despite the fact that \(R_{0}<<l_{em}\), an electron avalanche can develop inside such a bubston. Electron density should grow due to the increase in the energy of electron chaotic motion, stimulated by collisions with the bubble wall at the frequency \(\
u_{ew}=\langle v_{e}\rangle/R_{0}\) and due to the ionization of wall molecules (here \(\langle v_{e}\rangle\) is the mean arithmetical velocity of electrons) rather than due to their collisions with gas molecules (as is the case when \(R_{0}>>l_{em}\)). The ionization frequency is \(\
u_{i}\ =\ \dot{Y}_{e}/\Delta\), where \(\Delta\) is the ionization energy and \(\dot{Y}_{e}\) is the average rate of increasing energy \(Y_{e}\) of electron chaotic motion.