Thermal Nonlinear Klein–Gordon Equation for Nano-/Micro-Sized Metallic Particle–Attosecond Laser Pulse Interaction

In this study, a rigorous analytical solution to the thermal nonlinear Klein–Gordon equation in the Kozłowski version is provided. The Klein–Gordon heat equation is solved via the Zhukovsky “state-of-the-art” mathematical techniques. Our study can be regarded as an initial approximation of attosecond laser–particle interaction when the prevalent phenomenon is photon–electron interaction. The electrons interact with the laser beam, which means that the nucleus does not play a significant role in temperature distribution. The particle is supposed to be homogenous with respect to thermophysical properties. This theoretical approach could prove useful for the study of metallic nano-/micro-particles interacting with attosecond laser pulses. Specific applications for Au “nano” particles with a 50 nm radius and “micro” particles with 110, 130, 150, and 1000 nm radii under 100 attosecond laser pulse irradiation are considered. First, the cross-section is supposed to be proportional to the area of the particle, which is assumed to be a perfect sphere of radius R or a rotation ellipsoid. Second, the absorption coefficient is calculated using a semiclassical approach, taking into account the number of atoms per unit volume, the classical electron radius, the laser wavelength, and the atomic scattering factor (10 in case of Au), which cover all the basic aspects for the interaction between the attosecond laser and a nanoparticle. The model is applicable within the 100–2000 nm range. The main conclusion of the model is that for a range inferior to 1000 nm, a competition between ballistic and thermal phenomena occurs. For values in excess of 1000 nm, our study suggests that the thermal phenomena are dominant. Contrastingly, during the irradiation with fs pulses, this value is of the order of 100 nm. This theoretical model’s predictions could be soon confirmed with the new EU-ELI facilities in progress, which will generate pulses of 100 as at a 30 nm wavelength.