Figure 12 shows the dependence of the scattering coefficient on the electron density for χ = 1.5 in the asymptotic approximation and the Hankel and Bessel function series expansion approximation. One can clearly see a significant difference in the shape and position of the curves for different n when using the asymptotic approximation, while the approximation by series expansion has only inaccuracy in the form of a shift, which tends to zero with increasing order n.

Coefficients of spherical harmonics at βe = 0, χ = 1.5. The curves “exact” correspond to the exact values of the scattered field coefficients. (a) Asymptotic approximation. (b) Decomposition in series up to the first term.

In the general case, calculating the interaction of a high-intensity laser pulse with a group of dense spherical clusters located in three-dimensional space requires long and complex non-stationary calculations due to the fact that the electron density distribution of clusters as a result of interaction with the laser pulse changes over time.

To verify the magnitude of the electron density change in the case under consideration, we simulated the evolution of the electron density distribution in the one-dimensional space of a single cluster. The LPIC++ [8] code was used for the simulation. Two pulses were considered as sources: a primary frontal linearly polarized laser pulse with wavelength λL = 830 nm, duration τ, pulse amplitude envelope shape in time sin2(t), intensity IL = 1018 W/cm2 and transformed with wavelength λ10 = 83 nm, duration τ and intensity Ih = 1014 W/cm2. The period of laser radiation corresponding to the laser harmonic is T = λL/c ≈ 2.8 fs, so the pulse length in the simulation was taken τ = 10T = 28 fs, simulation time t = 20T = 56 fs. The plasma is represented by 2000 particles in each cell occupied by a target located in the center of a box of width wbox ≈ 3λ10; the electron density of the target in critical units is nel = 4.2nc. We also considered the interaction with the first harmonic, for which λL= 830 nm, IL = 1018 W/cm2. Single clusters of radius a from 9 to 60 nm were taken as targets.

Based on the obtained simulation results, the average total thickness of the transition layer during the interaction with the external pulse htr as a function of the target radius a was calculated (Fig. 13). The q-uasi‑stationary condition in this case takes the form htr ≪ 2a, which holds for a ≥ 20 nm. Thus, when using stationary calculations to estimate angular scattering, it is acceptable to use only clusters with a radius of 20 nm or greater.

Asymptotics of the average total thickness of the transition layer at 0 ≤ t ≤ 10T with respect to the radius of the target. nc used in the construction corresponds to the critical density for wavelength λ = λ10.